Unless the prompt to save a passkey looks like the screenshot below, then you are not saving the passkey in Bitwarden:
If you do not get a Bitwarden-branded pop-up (like the one shown above) when you try to create a passkey for some site, then there may either be a misconfiguration on your part (e.g., the domain may have inadvertently been added to the exclusion list under Settings > Excluded Domains), or your Bitwarden browser extension may have been locked or logged out, or there may be some incompatibility between Bitwarden’s passkey functionality and the website in question.
In such cases, you should not proceed with passkey creation if you do not want the passkeys to be stored outside of Bitwarden.