Yes, it is possible to set up face/fingerprint/PIN authentication across different browsers.
Face/fingerprint/PIN authentication is browser specific. For example, if you set up face/fingerprint/PIN sign in for Safari mobile then you can use face/fingerprint/PIN authentication only across this browser.
However, you are allowed to set up another face/fingerprint/PIN authentication across other browsers too. Hence you can set up multiple Face/fingerprint/PIN authentication for the same device across a different browser such as Chrome, Edge, etc.