What would Alexander's empire have been called?

[Subotan]

Actually, as far as I know, Alexander was different monarchs in the same person: he was King (basileus) of the Macedonians, Lord of Asia, shahanshah of Persia, Hegemon of Greece, etc.- but his uniting the rule of these different lands in himself was not the same as making all these lands into one united kingdom: there was not any such political entity as The Alexandrine Kingdom, or whatever you guys are trying to call it. This is one of many reasons why his conquests fragmented when he died: they had nothing to keep them together outside of his army and personality (the former more than the latter).

Don't project modern notions of statehood into an era where they have no meaning.

The Hapsburgs would have been proud

So that was more or less like the union of British and Scottish crown? (in this case, Makedonian and Persian crown, while they are still technically different nation?)

Nah, that was peaceful. Besides, Scotland was for all intents and purposes still an independent state until the Act of Union.