I’m not talking about “AES”, I’m talking about communism. By the definition of “communism = AES” then communism doesn’t abolish class, private property, the value-form, nations, etc.

Anarchism distinguishes itself from communism principally by an inherent opposition to hierarchy, and an opposition to many of the organisational forms that communists may advocate for or participate in, eg communist parties, councils, and any kind of structure that could constitute a hierarchy. And anarchists are inherently opposed to centralisation, and so on.