So the simplest answer is that Japan, Italy, and Germany were the countries that signed the Tripartite Pact at its inception in 1940, forming an alliance amongst themselves that became known as “The Axis.” This built on the alliance between Italy and Germany which had been known as the Rome-Berlin Axis.

Now to start getting a little more in-depth (though I’ll try not to get too technical). The question of which countries are meant by “The Axis” is a question of both the political and military ties between them, as well as the judgments of those outside of the Axis during its existence and when teaching the history afterwards. Romania and other countries like Hungary, Bulgaria, and Yugoslavia did eventually join the pact and there were other countries that allied with or became co-belligerents with the initial 3 Axis powers, but the reasons for not counting them among the “Axis powers” is a judgment call based on several factors.

One such factor is the agency of joining. That is: did the government of the country make a decision to join with the Axis? Now from a technical/legal standpoint the answer may be yes, but the circumstances surrounding the alliance may be such that we might consider it improper to say that the country joined of its own agency. A clear example would be Vichy France. Unlike Poland, from a legal standpoint France was de jure still an independent country under the dictatorship of Prime Minister Phillipe Pétain that aligned itself with the Germans. But de facto, France was conquered, it surrendered, and became a client state of Nazi Germany through the installation of a friendly ruler. So we don’t call France part of the Axis because whatever it says on paper, the French government did not, of its own free, willing, and voluntary agency join with the Nazis. Other countries in this boat would be Yugoslavia and Norway.

We also look at the exact nature of the relationship that the non-Axis powers had with Italy, Germany, and Japan. For example, let’s look at Finland. The government of Finland voluntarily joined the Nazis as what they considered co-belligerents against the USSR. Now the line between allies and co-belligerents is largely a matter of political philosophy, but several factors differentiate the Germany-Finland relationship with the Germany-Italy, Germany-France, or Germany-Norway relationships. Finland did not join the Tripartite pact, it did not integrate its army with the German army, and it didn’t really concern itself with any aspect of WWII as a whole (at least not in any major way), it had no expansionist goals of its own, nor did it share the political ideologies of Nazi Germany, Fascist Italy, or Imperial Japan. Rather, Finland was a liberal democracy, wished to stay a liberal democracy, but had a common enemy with Germany in the form of the USSR, which had waged the Winter War on Finland and took important territory from it between November 1939 and March 1940. Not to mention the history between Finland and the Russian Empire. So while Finland and Germany fought together, and the preexisting government of Finland did so voluntarily, the nature of the relationship was more “the enemy of my enemy is my friend” than a proper alliance.

However, after the Karelian Offensive, Finland surrendered to the Soviets and waged the Lapland War against the Germans. After the war, Finland was classified as a Nazi Ally in the Paris Peace Treaty and forced to pay reparations. But by the early 50’s those were paid and strange position of Finland during and after WWII made the narrative of Finland’s involvement in it subject to Cold War information campaigns. As a liberal democracy with rising economic power that also shared a border with the USSR, both sides of the Iron Curtain had an interest in holding diplomatic ties with Finland, while also shaming it if it got too friendly with the other. So the academic circles of both sides would throw around different analyses of Finland’s involvement depending on the political climate. Thus the politics of historical interpretation is certainly a factor, though it shouldn’t be overstated.

And with that all established, now we can get to Romania, which has its own unique story that you can say falls somewhere between those other examples. That is: It was a neutral country that became pro-Axis after a coup, which became pro-Allies after another coup and after the war became occupied by an Allied power leading to the rise of its own anti-monarchy Communist government. Whew you can already see the complications.

So the government Kingdom of Romania did not decide to willingly join with the Axis, but after it lost territory to the Soviets King Carol II was overthrown by an internal fascist movement (EDIT: I initially said this was done by a group called the Iron Guard, but the story is more complex than that. See u/RevAndroid’s comments below for more info). It was still a de jure kingdom under King Michael I (who had previously been king before Carol II) but was de facto a dictatorship under Ion Antonescu. Romania joined the Tripartite and became a staunch Nazi ally on the Eastern Front, and was responsible for a number of atrocities. However, the dictatorship was later overthrown by King Michael, Antonescu was ousted, and the Kingdom joined the Allies. In the Treaty of Paris, it got a similar treatment to Finland. It was recognized as an Ally of Nazi Germany and had debt for reparations imposed on it. It’s eastern territories were taken in by the Soviets, and the Soviet occupation of Romania proper helped bring about the Socialist Republic of Romania. So of course, the history became a subject of debate. The Soviets and Romanians couldn’t have this wonderful narrative of the rightful king being restored and bringing the country to the allied side. It’s too Tolkien-y and monarchist-y. They also didn’t want to mark Romania as fascist or Hitler-aligned as it was trying to maintain good diplomatic relations. And as for the West, well, let’s put it this way, the CIA was trying to support anti-communist forces in Romania and a lot of them were very much anti-semitic fascists, which still gets downplayed. So any narrative that got pushed was more along the lines of the positive story of the king and how he helped the Allies.

So in conclusion, the application of the term “Axis power” is generally just left to that simple reasoning of the initial signers of the Tripartite Agreement. Whatever the military size and role that a country had, Germany, Italy, and Japan were the three countries that were really exerting their own power and ideas on the world, and other countries aligned with them, like Romania, have more complicated or insularly self-serving roles in WWII. And the 75 years that have passed since the war ended were largely filled with further divisions that politicized the very teaching of these countries’ roles in WWII. Today, the roles those countries played may not be widely suppressed or hidden, though I’m sure you can find plenty of debate and politicized history if you dig for it especially within those countries. But for the purposes of general education of WWII outside of those countries, especially in grade school or secondary school, teaching the Axis as the Big Three is often enough.