You keep your UUID as a backup. The fingerprint system on my phone still has an option to log in with password, the fingerprint is just a convenient, faster alternative. It's the same here, you keep the UUID in a folder with other sensitive documents and when you lose your finger, you fish it out, log in with it, and register another finger.
You'll have to remember multiple UUIDs, one per service. Now that I think of it will be rather cumbersome. And bio data has to be editable if it will be usable at all i.e. add/delete new entries - fingers, eyes etc. Damn, it is harder than it seems. Anyway, bio data as a login should be implemented, we only need to think exactly how.
You have to remember your UUID? Probably it would be more like a file than a string.