Institutional arrangement aren't irrelevant and cause parallel political struggles to play out differently. For example, the story of abortion law in the U.S. has been one of legislatures imposing limits and courts striking down those limits, while in much of Europe it has been one of legislatures legalizing abortion and of courts imposing restrictions.
But, ultimately, both the elite and grass roots political culture of a country has proved to be more important than institutional arrangements.
For example, empirically, western style democratic systems generally fare very poorly in places where there is a high level of cousin marriage, even though nothing in the theoretical structure of western style democracies obviously assumes any particular level of marriage to closely related people.
U.S. style three branch systems
The tripartite division of the United States government into an executive, legislative and judicial branch was widely copied by U.S. states and in the initial constitutions of Latin American countries (most of which have been amended many times over since then) because it was the first democracy that functioned as a republic for a prolonged period of time. The tripartite division was based upon a political theory developed in 1748 and political science has learned a lot from subsequent experience with democracy which was mostly a hypothetical concept at the time.
Moreover, while many Latin American political systems started with formal constitutions almost identical to that of the U.S. political system evolved in practice has been profoundly different from the U.S. experience. (A thin summary without much context can be found here).
Unified parliamentary system
But, the British model in which the executive and legislative branches are fused, the upper house of parliament has little real power, and the judiciary does not have the power of judicial review (indeed, in the U.K. for most of its history, the highest judicial body was part of parliament), can work well.
The fused parliamentary model has not proved to be notably inferior and has been widely copied outside of Latin America. For example, few people would say that Canada's British style system is clearly inferior institutionally to the U.S. system.
One party states
One party states on the Soviet model, in which the formal political institutions are more or less irrelevant, have also proved very popular. These states have multiple branches of government, in theory, but, in practice, all branches of the government are controlled by the Communist Party.
Most new western style democracies fail
It has turned out in reality that neither a U.S. style constitution, nor a British style parliamentary system has been stable enough to maintain in the absence of the corps of experienced politicians, senior civil servants, lawyers and judges necessary to make that work and a grass roots political culture equipped to play its role in the process. Almost every newly independent country that did not have long experience with Western style democracy has seen that system collapse early on, into either a military dictatorship following a coup, or a one party state on the Soviet model.
Some of the most recently failed or currently failing new Western style democracies are South Sudan and Egypt.
This is not just limited to the Third World.
Russia's original one party system was a result of the swift collapse of a Western style democratic system. South Korea, Spain, Portugal and Greece all spent time in the 20th century under military regimes following coups. Germany's Western style democracy under the Weimar government swiftly collapsed into an authoritarian, single party Nazi regime. Mexico had a dominant party system that was almost a one party regime for two generations. Further back in history, the original electoral democracies in France and Britain originally failed and reverted to absolute monarchies before re-emerging as democracies on later attempts.
In contrast, the one party states that emerged in Eastern Europe, in Iraq, Syria, Egypt and Lebanon, in most of sub-Saharan Africa and much of Asia, were reasonably stable, often escaped the hereditary principle, and allowed for a modicum of public order and economic development, even if neither was as robust as in Western style systems.
Successful western style democratic transitions
The successful transitions to Western style democracy have usually involved a sustained period of self-rule in a colonial arrangement (e.g. the U.S., India, Canada, Australia, some Caribbean and Pacific islands, Iceland), or a supportive monarch who remains in power and legitimate but voluntarily cedes power (e.g. Thailand and Japan). The rare exception to these two models, East Timor, was a case that involved extensive and prolonged hand holding and state building by the U.N. in a manner similar to an extended period of colonial weaning.
There are also successful western style democracies that emerge after prolonged period of military or one party rule as the political culture gradually develops such as Turkey, Indonesia and Mexico.
In states with "dual executives" there can effectively be four or more branches of government that are largely independent, a judiciary, a legislature, a parliamentary cabinet, an Presidency with real power, and sometimes an effectively autonomous civil service.
Russia and France developed hybrids of the American and British style constitutions, with a parliamentary model legislature quite unlike that of the U.S. Congress, but a Presidency that is stronger than that of parliamentary countries like Israel where the President merely serves the role of a symbolic leader akin to a constitutional monarch. These are sometimes called "dual executive" or "semi-presidential" states.
In France, moreover, the civil service is so powerful and independent that it is sometimes considered a branch of government itself leading to a "triple executive." The civil service also contains is own quasi-judicial system known as the "Council of State" which handles matters of administrative and public law (as opposed to the criminal cases and private law cases handled by ordinary courts), in what is widely considered one of the most effective and functional public law judicial systems in the world. So, France arguably has six branches of government (the ordinary judiciary, the Council of State, the executive functions of the Civil Service, the legislature, the cabinet, and the President with real executive authority).
The European Union also has a highly elaborate multi-branch internal structure, a design no doubt conceived by someone with French institutional sensibilities (as France was one of the original European Community members).
As an aside, it is also worth noting that many countries classify prosecuting attorneys as part of the judicial branch rather than part of the executive branch as the United States does.
Even in the U.S. government, independent agencies, such as the U.S. Postal Service, the Tennessee Valley Authority, the Securities and Exchange Commission, the National Labor Relations Board, and the Federal Reserve effectively constitute a fourth branch of government not contemplated when the U.S. Constitution was adopted with significant regulatory authority and a lack of a direct reporting relationship to the President in the executive branch, even though the people who run these agencies are often appointed by the President with U.S. Senate approval to fixed terms of office (the Federal Reserve system is even more insulated from the President with member banks who report to private shareholders having a significant say in its governance independent of the President, although key regulatory authority on the Board of Governors for the entire system does require a Presidential nomination and U.S. Senate approval).
Many U.S. states similarly have multiple independently elected figures in the executive branch (for example, in Colorado, a Governor, Lieutenant Governor, Secretary of State, Attorney General, State Treasurer, independent state school board, and independent state board of regents for public colleges) as a product of progressive era reforms (for example in Washington State see here). (A broader discussion of historical progressive era reforms aimed at expanding direct democracy can be founder here). So, state governments often effectively have more than three branches. From a political perspective, the independence of a state attorney general, which effectively puts in place a permanent independent prosecutor and independent chief regulatory officer for the state with wide discretion in political matters, is particularly important when one is conceptualizing an arguable independent branch of government, even though the AG usually as the Governor as his or her primary reporting relationship with his or her client, the State. Many of the other statewide elected officials in the executive branch (e.g. the State Treasurer) typically have far more ministerial roles that limits their independence from the rest of the executive branch and limits their meaningful political power. Also, as in the federal government, independent agencies at the state level effectively constitute an additional branch of government and this is particularly true, for example, in the case of public college systems with independently elected boards of directors.
Unique systems of government
There are also a couple of governmental systems that don't fit neatly into boxes.
Iran is a theocracy, but its religious leaders have allowed a fairly robust democratic system with genuine elections for President and legislative leaders to take place.
It isn't a full Western style democracy because religious leaders can and do disqualify candidates running for office and the state is ultimately subordinate to them, and religious leaders also censor public discussion, but there are still competing candidates within the realm of what is acceptable to the religious leaders who run genuine election campaigns that are resolved by popular vote with a wide franchise and significant public participation. In this system, the Islamic authorities constitute a fourth branch of government in some ways comparable to a President with real authority, or a monarch in the transition period between an absolute monarchy and a purely symbolic constitutional monarchy (a bit like the situation that prevailed in England around the time of the American Revolution).
Almost all of the new democratic constitution in the Islamic world, such as those of Afghanistan, Iraq and Egypt, have sought to follow the Iranian model of a Western style democratic arrangement of some sort which is subordinate to Islamic law, although these countries don't generally have an entire religious branch of government the way that Iran does on an institutional basis.
In Saudi Arabia, there is no effective independent judiciary, and power is split mostly between the monarchy, the civil service responsive to the monarchy, and the religious authorities. It is a poor fit for the three branch model of government.
Other absolute monarchies (most of which are Islamic these days) don't have such a clear delineation between royal and religious power and so are effectively one branch, rather than two branch systems. Some of these Islamic monarchies do have legislatures, but they are effectively impotent.
The other notable absolute monarchy is the Roman Catholic church which also rules as a secular government in Vatican city. Its organization is beyond the scope of this post.
China's Post-Soviet single party state with local non-partisan democracy
China started out as a pure one party Soviet style state, but has developed on its own path, has free and fair non-partisan local government elections in much of the country, and has party institutions that are far more responsive to public opinion that Soviet style one party systems ever were, albeit with highly government censored speech, no free and fair elections with a wide franchise, and limited political and economic rights despite an emerging system of private property and an emerging market based economy.
China, while it may have institutions that can be labelled as executive, legislative or judicial, effectively fuses the three functions in a common chain of command following precedents associated with the Confucian prescription for the organization of government which focuses more on finding good rulers in a rule of man, than on finding good rules that bind the rulers in a rule of law. Its citizens may lack liberty, but it is hard to argue with a system that is lifting a large share of the world's extremely poor people out of poverty and experiencing double digit economic growth every year for decades.
Systems With Permanently Brokers Power Sharing Between Factions
Lebanon has a Western style democracy, but imposed on that is a long standing power sharing agreement that gives each of its major ethnic factions a particular senior post in the government, so that each has a stake in the system. Effectively, each ethnic faction has an incomplete state within a state that it can run in cooperation with the other factions.
Iraq's post-Iraq War constitution sought to emulate the Lebanese model, but this has been a failure giving rise to a disaffected Sunni Arab faction as non-Kurdish Shiites became dominant, with Kurds largely turning to self-government in a fragile alliance with the national government and Sunni Arabs eventually not seriously resisting succession into an Islamic State.
New Zealand is a less extreme variation on power sharing. It has a highly unitary British style parliamentary system, but does guarantee its indigenous Maori minority population some designated seats in parliament elected by Maoris alone, rather than by the general population. Also notably, in New Zealand, until 2004 when it created its own Supreme Court, its highest judicial body was the Privy Council of the House of Lords in London, so ultimate judicial authority (which in principle could overrule or interpret decisions of its domestic legislature) was vested in another country rather than being part of its own political system in a last vestige of its colonial relationship with the U.K. (as well as the fact that it shared and still does share a monarch with England who rules in New Zealand via a governor-general whom she appoints). The governor-general's office and privy council could be considered additional branches of government.
Unlike most constitutional monarchies, Malaysia has a number of regional monarchs called Sultans in nine of its thirteen states who rotate in the position of king of the entire country (a bit like the European commission of the European Union), and these system on shared monarchy effectively constitutes its own branch of government.