Why do I see the term 'oligarch' almost exclusively applied to Eastern European countries? Are there oligarchs in the European Union, the United States? As I understand it, an oligarch is a wealthy businessman with significant political clout. It doesn't seem cogent to argue that there are no people in America who meet those criteria (not sure about the EU).
Technically speaking, an oligarch is a member of an oligarchy: a system in which political governance is given to a cadre of wealthy, propertied individuals in society. The archetypal example is the Republic of Venice, which lasted from the 7th century to the 18th, and was ruled by a council of businessmen and aristocrats. We could also look at Athenian democracy, which — despite its reputation as egalitarian — only allowed settled property owners to participate as citizens, excluding laborers, women, immigrants, and others. There are no true oligarchies in the modern world that I am aware of. Russia currently has a constitutional democratic republic, not too dissimilar from the US system, so there are no 'oligarchs' there in the proper sense.
The term 'oligarch' is used more loosely to indicate a person who is exceedingly wealthy, and uses his wealth and influence to pressure or coerce governments into producing favorable policies or legislation. Russia is particularly prone to this, because as a nation it is new to both democratic politics and capitalistic practices. That leaves it vulnerable to corruption, crony capitalism, economic blackmail, and other conditions that give the ultra-wealthy an excessive say in government decisions. But by that definition there are 'oligarchs' practically everywhere. In the US alone we can point at Rupert Murdoch, the Koch Brothers, Dan Cathy, Warren Buffett, George Soros, Mark Zuckerberg, Mike Bloomberg, Donald Trump, most of the power-players on Wall Street and in the financial industry, and others who have used their wealth and influence to try to bend (some might say pervert) the course of US politics.
In the US we hear a lot of talk about 'Russian oligarchs' because 'oligarch' is generally used as a term of derision, and it is an easy way for pundits to express their distaste with some Russian activity, and with any American opponents who might have dealt with these people. We do not often hear about 'American oligarchs' because the people we would label as such — those whose behavior is more or less equivalent to their Russian counterparts — are in positions of power in this nation that make such labels tricky. Would any of Murdoch's or Bloomberg's news media be inclined to name their corporate owners as oligarchs? Would any politician (aside, perhaps, from Bernie Sanders) be willing to paste that label on a mega-donor whose money and influence might be critical in an upcoming election? Set aside that this type of trepidation is precisely what we would expect in a system with oligarchs — that is how an oligarch of this sort throws his weight — it is unlikely that the label will come into common use for such people.
Are there oligarchs in Western countries?
No. Though it might depend which countries you consider "western" - it is a vague term.
A definition of oligarchy is:
a small group of people having control of a country or organization.
However in relation to "Eastern European" countries, the term is usually applied to countries where the government of the country has been captured, and exclusively run by a small group of people who have enriched themselves by appropriation of state assets and powers, by confiscation of private businesses owned by non-group members and by misuse of state powers (police, judiciary etc) to establish their own private monopolies.
In general, political power in western countries is not restricted to one relatively coherent small unchanging established group of individuals.
Typically "Western" countries see a change, or credible possibility of change, in power between distinct groups (or differing coalitions) about every five years.
You can argue that some wealthy individuals in the US or Western Europe have undue influence. Though undesirable, and against general European or US democratic principles, this is not the same thing as an oligarchy.
It can be argued that, in one or two "western" countries, there is some apparent malfunctioning of the usual mechanisms for removing from office officials who are seen to be corruptly misusing their office for personal gain. This has not yet reached the stage where those countries can be clearly regarded as oligarchies fully and irreversibly captured by a small closed group.
This is debatable, of course, depending what precise definition one attaches to "oligarch". If you consider a stricter one, RedGrittyBrick's answer is essentially correct.
On the other hand, Berlusconi has been called "Italy's oligarch". Wikipedia's page on Berlusconi cites a Foreign Policy article titled "Now the Czechs Have an Oligarch Problem, Too" which opens with
Czech oligarch Andrej Babis, his country’s second-richest man and one of the most politically powerful billionaires in the world [...] The rise of Babis — nicknamed “Babisconi,” [...]
A 2009 article in Bruegel says
Rich Western nations have long lived in the comfort that government capture by specific private interests was mostly a feature of poor or emerging countries, rather than of themselves. The expression “crony capitalism” referred to Asian countries in the late 1990s, and “oligarchs” sounded as if they were unique to Russia. But the crisis has shattered the West’s sense of relative comfort on this front. A telling indication was the wide impact and debate raised by an article published by the Atlantic magazine in May. Its author, Simon Johnson, a former chief economist at the International Monetary Fund, claims that the US has become prey to a financial oligarchy, which played a key role in both creating the crisis and preventing appropriate policy responses to it. [...]
In Europe, business and financial oligarchies are prominent as well, and their responsibility in the crisis is hard to deny. In the United Kingdom, the City of London largely shaped the now discredited “light-touch” regulatory approach of the Blair-Brown years. Next door, a government bail-out in February elicited the dry comment by the Financial Times’ Lex Column that “the Irish bank executive self-preservation society has scored another success”. In Germany, political and financial elites are interlinked to the point of fusion in much of the banking system, and the epitome of this relationship, the local-government-owned Landesbanken have been among the worst cases of poor risk management. A comparable situation exists in Spain with the savings banks that fuelled a monumental real estate bubble. In France, a few wealthy individuals control most of the media, and the heads of leading financial firms have been reported as providing influential policy advice throughout the crisis. In Italy, the Prime Minister ranks among the country’s richest individuals. And so on.
After discussing the density variation of billionaires among countries it says
Oligarchic power also highly depends on size, as Adam Posen, an American economist, had noted just before the crisis: in a small country or regional government, the wealthy few can more easily control the scene than in a large and diverse system (on the other hand, decentralization lowers the risks of bureaucratic capture or of policymakers’ estrangement from local realities). From this standpoint, EU institutions may be less prone to oligarchic capture than individual member states. Lobbying is certainly more conspicuous in Brussels than in many national capitals, but partly because it is more transparent -though still insufficiently so. Business elites are generally too diverse in Europe to form a single social group, which comparatively limits their ability to direct the policy process. National oligarchies often attempt to influence EU-level decisions indirectly through national governments, especially of large countries, rather than directly in Brussels.
So the term does get a looser meaning the further left on the political spectrum (of the speaker) you move... and apparently also (depending on) how close you are to a financial crisis.
Individuals such as Sheldon Adelson, David and Charles Koch, and the late Harold Simmons and a group of wealthy donors assembled by Republican strategist Karl Rove felt that they needed to speak out to ensure that the country had stronger leadership and moved in what they considered a better direction. But they were not the only super-wealthy people who were politically active. In recent elections, there has been an explosion of activism by the rich. Billionaires such as Michael Bloomberg, George Soros, and Tom Steyer have poured extensive resources into supporting their favored candidates and causes. In addition, wealthy individuals have bankrolled advocacy campaigns at the state level—for example, in support of same-sex marriage and marijuana legalization or in opposition to Obama’s health care reform and higher taxes on the wealthy. Aided by friendly Supreme Court rulings and the rising cost of election campaigns, affluent people have discovered that they are in a strong position to affect a variety of different issues.
In researching this subject, I discovered that it is not just an American development but something that is happening globally. Billionaires have run for office in Austria, Australia, France, Georgia, India, Italy, Lebanon, the Philippines, Russia, Thailand, Ukraine, and the United Kingdom as well as the United States. Most of them have won. Oligarchs in Russia, so-called “princelings” in China, and tycoons in many other countries are becoming politically active and affecting public policy. Their political involvement raises important questions about excessive influence, especially in places where there is weak rule of law, overt corruption, and limited opportunities for social or economic advancement. The activism of the super rich is taking place against a backdrop of poor transparency, weak news coverage, accountability problems, and performance challenges on many different fronts in political systems around the world. With the “wealthification” of politics, those in the upper echelon, who as a group hold policy views that differ significantly from those of the general population, have access to many ways to influence the political process.
But even in there, there's a reluctance to apply the term "oligarchs" to a US context; instead one complains about the political influence of billionaires.
There is a spectrum of oligarchy which is defined by distribution of wealth and despotism/autocracy. Oligarchs completely control police, military, banks, newspapers, television and law courts. Western elites also control those things, only partially and secretly. They are LESS above the law than oligarchs. Oligarchs dictate the law, western elites lubricate the law.
Western elites struggle to even chop up a journalist and fly him home in pieces, or give a dissident coffee mixed with radionuclide polonium-210 or botulinum. The west holds fewer political prisoners for "incitement to trouble to the public order", about 1-10% as many ideological prisoners.
Free press and independent judiciary detect who trade enriches and by how much, so it is harder to hide money within a nation while doing business there. For that reason, the west doesn't have extreme and overt oligarchy issues, the west has partial and intricate control of governments by the wealthy and by corporations, which follows different rules than an oligarchy and is better described by different terms, like lobbygarchy and donorgarchy.
Eight western nations own about 50% of the world's trade and financial assets, they have a lot of control, and it's held by a broad association of groups and elites rather than a few families.
In western countries any oligarchic parallels have to be hidden and discreet, and oligarchs have very good friends in western banks, they can buy influence, but they can't do so overtly, for example the Spanish king who allegedly received 100 million from Saudi Arabia for influence, had to hide his friendship with the oligarchy.
The main form of financial control of western governments comes from multinational corporations AKA "multinationals"... Corporate groups valued in trillions that distribute operations all over the world and which can move money from one nation to another.
The following graph from 2014 by Credit Suisse, a Swiss bank, plots "known wealth" while Switzerland at the time was known to be an opaque tax haven prior to the Mossack-Fonseca scandal... The green percentage lines max out at 22% of national wealth owned by the richest millionth.
The term oligarch does not mean quite what you (or the people writing those articles) think it does. An oligarch is a member of an oligarchy, which is a system where political power resides in the hands of a small number of people. These people may not be wealthy (at least to start with, though human nature being what it is, they tend to accumulate wealth). For instance, the upper power structure in the USSR prior to Stalin might have been considered an oligarchy.
The reason that Eastern European countries are sometimes thought of as oligarchies is that there are relatively few wealthy people, these people generally obtained their wealth at least partly through government connections, and they tend to form a clique.
In the US there are lot of wealthy people - over 600 billionaires and 18.6 million millionaires, per Google (though that's probably prior to the current market crash :-(). Those people have diverse interests and opinions, so they don't form the clique necessary for an oligarchy.