I've just noticed that one of the SQL servers doesn't do application aware processing even though AAP works for the other servers of that customer. So I went and hit test and for that machine, I receive the error message in the subject.
Google brought me to this issue on vSphere, where it is resolved by reinstalling/updating VMware tools. I would have expected that it was the same on Hyper-V with integration tools, but Veeam is not complaining about them.
Is there any other thing I have to check?