Ok, today I exchanged the unit at the store. The new one shows the same behavior. M newest insights (tl;dr: no solution found):
- It works with the wifi at my parent's home (IIRC a Speedport W 701V router with b+g standard on channel 2 with WPA/WPA2 PSK)
- tried to emulate the same settings with my Fritz!Box and it didn't work, even without any encryption. I'm puzzled...