A fingerprint is like a key that ties your Machine to your alias (machine@ad).
It is saved in the file %programdata%\AnyDesk\service.conf. You can make a backup copy of this file and restore it later to keep your alias and fingerprint after events such as re-installation of the operating system or pc upgrades.
Please keep in mind that anyone who gains access to this key file can set up his machine so it looks like it is your machine.
For security reasons, it is not possible to re-acquire an alias whose fingerprint has been lost. This prevents identity theft.

If you obtain a licence for AnyDesk, you can reserve your own namespace. This makes it possible to change your alias freely.