There are two people in the room named Chris.

I assume you want to talk to Chris, both of them.

I assume you want to talk to Chris (both of them).

1) Both are grammatical sentences, but how would you label the phrases in bold?

2) Can you really call it an adjective phrase when it doesn't make sense when made into a relative clause?

"..Chris, who is both of them."

3) But what else could it be? Are there some phrases that just can't be classified into a phrase type, perhaps?

Actually, I don't understand the sentence. Is it an appositive phrase claryfying the noun Chris, or is the sentence intended to be read otherwise?
No tricky, second meaning. Simply meant to be saying that both of them are included.