No, because the phisher will get a token that is designated for, say, mircos0ft.com which microsoft.com will not accept. It is signed with the user's private key and the attacker cannot forge a signature without it.

I think we're talking past each other a bit here.

If I were trying to phish someone, I wouldn't attack the public key crypto part, so how domains come into play during authentication doesn't matter. I'd just grab the "unencrypted" session token at the end of the exchange.

Even if you somehow protected the session token (sounds dubious), there's still plenty a phisher could do, since it has full MITM capability.

A password manager is also not going to fill in the password on mircos0ft.com so is perfectly safe in this scenario. You need a MitM-style attack or a full on client compromise in both cases, which are vulnerable to session cookie exfiltration or just remote control of your session no matter the authentication method.