Just to close this topic and give you some feedback in case somebody is interested:
I have got back my uniti two weeks ago now. The naim support could not really reproduce the faulty behaviour, but I have the feeling that they had noticed some other issues maybe. Anyway, they have decided to exchange the new v3 streaming board with another one. With this new board the uniti does play without any issues again in my home network. Great! I am so happy!
(I know, this still does not prove anything. Maybe my fritzbox router did come to hate the first instance of the streaming board but works friendly together with the new one!?)
A big thank you to the naim support! It took some weeks to get my uniti back, but the whole investigation and hardware exchange was done on warranty, as I had upgraded the uniti 1 only some months ago to the new streaming board.