Armenia, for all intents and purposes, is considered geographically European. The same is true of Georgia. Even Azerbaijan is included in European leagues. This is not due, as far as I know, to any form of political judgment. If it were, it would make far more sense to put Azerbaijan separate from Armenia, or from Turkey. Armenia, Georgia, and Azerbaijan are simply in a geographically awkward area, at the intersection of the Middle East, Asia, and Europe. As such, they defaulted towards Europe, while states south and/or east of them are considered Asia and/or the Middle East (i.e. Iran, Syria, Turkmenistan, etc.).

However, you have touched on an interesting exception outside of this: Israel. If Lebanon is in the Asian group for FIFA soccer, and the Asian group for FIBA Basketball, why is Israel in the European groups?

Even more puzzling, why is Lebanon in the Asia-Pacific group in the United Nations, while Israel is in the Western European Group as of 2000?

The main reason is politics here. Quite frankly, Israel is treated specially because Arab states refuse to interact with it, believing that constitutes recognition, even in sports.

Perhaps the best illustration of this is in sports itself, in the case of soccer (football, for the rest of the world!). Due to the Arab League refusing to interact with Israel, numerous Arab states would regularly forfeit games when they came up against Israel. The most absurd iteration of this came up in 1958, when Israel reached the qualifying stage for the World Cup without playing a single game. Every opponent forfeited rather than play Israel, though FIFA intervened and forced Israel to play a game against Wales to qualify, which Israel lost. In 1974, the Muslim world pressured FIFA to allow Israel to move to Oceania, and in 1991 it joined Europe's league.

A similar case explains Israel's participation in the United Nations. Israel was finally moved in 2000 (this skirts the 20 year rule, but I think it's sufficiently historical and close to fit), after 39 years of being part of no regional group. This meant it was significantly disadvantaged, and could not hold a variety of UN positions. Israel was the only country denied membership in a regional group it clearly fit, to my knowledge, while still being a UN member. This meant it could not be elected, for example, to a seat on the UN Security Council until it joined a group, and it was forced eventually to apply to the Western European group when invited in 2000, after 39 years of no representation in the most powerful UN body. This is, again, due to the refusal of Arab states to recognize or acknowledge Israel's existence. Even then, the acceptance to a group was conditional (that has evolved, but I'm not going any further into the 20 year rule!), meaning it still couldn't join the Security Council, but some other bodies could still be participated in.

Israel is, in short, treated uniquely due to the discriminating actions (I don't mean this as a charged term, I mean it in the literal sense, the Arab world targets Israel for special negative treatment) of the Arab and Muslim states in various regional bodies, which often form the linchpin of international systems (including in sports). As a result, other regional groups have historically chosen to accept Israel, sometimes under pressure from Israeli allies (for example, the US heavily pressured Europe to invite Israel in 2000 in the UN) and it has ended up being in separate groupings from most of the states it borders.

Armenia and Georgia, on the other hand, do not typically have such political considerations as far as I'm aware.

I should clarify two things:

1. Israel is the only country that was ever barred from a regional group despite attempting to enter it. This contrasts with the United States, which has chosen voluntarily to avoid regional group membership but retains its permanent seat on the UN Security Council, and is part of the Western European group for electoral purposes (again, all by choice).

2. Other states have, in the past 20 years, faced restrictions on regional group membership, but the situations are complex and in my view not entirely equivalent, but I can't discuss them in depth here.