If you're copying passwords out of a password manager and pasting them into password fields, then yes, you're getting a significant improvement to phishing protection with a hardware key. If you're using the password manager's autofill feature, and that autofill feature is bug-free, then you're not getting any additional phishing protection.

Your passwords can still be stolen, however. Any hardware authentication mechanism is going to ensure that no matter how compromised your local machine is, the worst an attacker can do is steal one active session. They can't steal the secret required to initiate any future session.