Okay, so now I understand the problem:
Turns out I had the following option enabled:
Disable DHCPv6 server in the FRITZ!Box:
- Enable the M and the O flags in the router advertisement messages of the FRITZ!Box (SLAAC possible)
I only run DHCPv4 on the network, I rely on SLAAC for IPv6 (and manually setting IP addresses for devices where I care about static IPs). So it was incorrect to send the M and the O flags, as they mean that devices should look for DHCPv6 service. Other devices are fine with that, but apparently Fairphone is not.
Once I changed the config on the router to:
Disable DHCPv6 server in the FRITZ!Box:
- There are no other DHCPv6 servers in the home network
everything works fine. The IPv6 works (including on the phone), and WiFi is stable now.
Thank you all for hints.