It is technically possible since it has been done.
As the user registers with or for a new account, you also enroll one or more fingerprints. To do that, a fingerprint scanner is required. (Note that the fingerprint
image that the Android fingerprint reader captures, cannot be "taken" out of the phone. You need an external fingerprint reader.)
Also needed is some sort of SDK for extracting a template from the fingerprint image. (The template is close to what you might be calling the unique identifier, although it might not really be that unique.) The manufacturer of the fingerprint reader, usually also supplies the SDK. If they do not, you'll need a third-party SDK. One can search online for those.
The extracted template is stored on the server alongside other user data. At login, you extract a template from the same finger, transmit it to the server, and search amongst the previously enrolled templates. For that search (identification), you also need some SDK, or some sort of AFIS. There are commercial as well as free offerings (such as sourceAfis).
A better explanation is given in the Wikipedia article.