I use Bitwarden in Firefox and passkeys "just work" on Linux, Android, Mac, and Windows. Previously, I used the extension in Chrome (Linux, Android, Windows).

The only relevant Bitwarden setting appears to be "Ask to save and use passkeys" under Notifications. I do turn off the browser's built-in password manager, though I believe anything else relevant I have at default. If you have those and they still aren't getting saved, then I'm at a loss, but wish I knew why they don't work for you. In Bitwarden, you can see if there's a passkey saved in an entry, as the creation timestamp is shown right under the password field and editing an entry also allows deleting a passkey.