Both the U.S. and U.K. Have Regional Teams
Both the United States and the United Kingdom actually have regional football (soccer) teams. In the U.K., there are teams for major regions of the U.K., including England, Wales, Scotland and Northern Ireland. In the U.S., these soccer teams are usually centered in various major cities. Typically there is only one team per state, although some bigger states have two teams. A similar pattern emerges for American football, baseball, basketball, hockey and other sports.
Very Different Kinds of Unions
Both the United Kingdom and the United States are unions, but their legal structure is considerably different.
The U.K. is a Unitary State and Constitutional Monarchy
In the case of the United Kingdom, it is structured as a unitary state, with all power emanating from the central government. The central government can delegate power to its political subdivisions, but it also can take that power away at any time. The central government is the only sovereign in the United Kingdom, and everything else is considered a political subdivision. Political posturing aside, it is Westminster that holds ultimate power.
The U.S. is a Federal Republic of Sovereign States
Contrast this with the United States, which was formed through the ratification of several agreements by 13 independent countries that each had its own government, its own military, its own currencies and its own laws. Each state considered itself an independent nation, similar to how European Union member states view themselves as independent nations.
Because the states considered themselves independent countries and sovereigns of their own, they entered negotiations with that mindset, similar to how EU member states entered negotiations to form the European Union.
The result was the Articles of Confederation and later the U.S. Constitution, which divided power between the federal government and the states, with both being sovereigns under the new federal system.
Under the agreement (the Constitution), neither party (the federal government or the states) are allowed to break the terms of the agreement. If the states break the agreement, the federal government will intervene. If the federal government breaks the agreement, many of the states will attempt to leave. This model of duel sovereigns is actually an important check and balance against tyranny, since both sovereigns have the power to enforce the agreement.
Changes in the U.S. Power Dynamic Over Time
Originally the federal government was only allowed to handle foreign affairs and interstate commence issues, plus other items specifically mentioned in the Constitution. All domestic issues were left to the states. And until the civil war, a majority of domestic law making power resided with the states, not the federal government.
Things like roads and highways, law enforcement, health care, welfare, public works, insurance and banking regulations, driving laws, etc. are all domains of the states. Each state still, to this day, has their own state military for defensive purposes, in addition the the federal government having their own military. Things like murder and running a red light are against state law, not federal law. States actually have to use extradition procedures to transfer criminals from one jurisdiction to another.
Before the civil war, the states had most of the power. This changed in a couple important ways after the civil war.
First, the federal government started enforcing civil rights violations after passing several amendments to the Constitution. This gave the federal government new powers to control state laws that violate citizen's rights (which is a good thing).
Secondly, the federal government found a loophole in the Constitution. While it is unconstitutional and illegal for the federal government to make laws related to domestic issues (such as health care, education, law enforcement, roads & highways, etc.), they can decide how they allocate federal funds. So basically, as an example, they are not allowed to make a national speed limit of 55 mph (88 km/h), but they can withhold federal funding if the state does not pass a state law making the speed limit 55 mph (88 km/h). (This actually happened and the Supreme Court actually ruled on this.)
The result is the states often bend to the will of the federal government to get the federal funds, but have the power to refuse the money and do what they want. Richer states obviously have more choices than poorer states that rely on federal funds.
So you wind up with weird things like a state passing a law because the federal government wants it, but then not enforcing it or making the penalty small. Or some states adopting a federal law, and others rejecting it. A constant tug of war of power and money between the states and federal governments.
So, in summary, the United States is a union of sovereign states that have the power to create and enforce their own laws without approval of the federal government, subject only to the Constitution of the United States of America and their own state constitution.
Whereas the United Kingdom is a unitary state that sometimes delegated powers to various political subdivisions, either for efficiency or for political reasons.