- His dog is his best friend.
turns out to mean the same thing when the two NPs -- his dog and his best friend -- are reversed:
- His best friend is his dog.
These are sometimes called "equational" sentences, because they're commutative, like an equation.
This leaves the impression that is means 'equals', which appears true, but only in this construction.
In fact, both Noun Phrases refer to the same individual, described in two different ways,
and both NPs can be predicate nouns. That's the key to equational sentences.
In the first sentence, his dog is the subject NP and his best friend is the predicate NP.
In the second, where the nouns are reversed, the roles are reversed as well --
his best friend is the subject NP, while his dog is the predicate NP.
In both cases, is does not mean "equals" -- it doesn't mean anything at all.
It's just the auxiliary be that's required with any predicate noun or adjective:
- His dog is tired. His dog is a dachshund. His dog is his best friend.
Calling any of these sentences a "copula sentence" is wrong, because that term presupposes that the verb be is being used with a special ("copular") meaning here, while in fact it's just an ordinary auxiliary verb, governed entirely by automatic syntax, and totally meaningless otherwise.