The team seems to have found the regression. It's upstream, and also applies to Chrome. It's being worked on as we speak to see whether the functionality can be restored, and whether existing saved passwords can re-appear.
It seems the built-in password manager functionality might be moving to Play Services in a few months, and Google started the migration to that from now, which seems to be what's causing the regression here.
We'll update everyone when we know more.
Thank you all for your patience.