@Tom I recognized, that AnyDesk does not update itself when there is a new version available.
In the AnyDesk Help Center you can read:

AnyDesk versions on the release channel are normally moved to the automatic update channel a few weeks after release following feedback from our users.
Source: https://support.anydesk.com/knowledge/update

So the new version is automatically updated a few weeks after release.
In the case of my brothers PC, the new version was not even installed after several months (he still had an older version and Auto-Update did not download the current version).

So an Auto-Patch of AnyDesk would be very helpful.