Can not reconnect after external IP-change.

Please read carefully.

I helped someone out and have setup an old PC running debian 8 (Jessie).
The server-machine is behind nat in a normal home-setup.
NO-IP is used to take care of the IP-changes and that works fine.

TS3-server is started on bootup using systemd, this works fine too.
I have sshd running there so I can manage that PC from my home, working great.

Now the following happens:

Her ISP gives a new external IP (every day) and then my connection via ssh drops and also the TS-client disconnects.
This is fine and to be expected.
However:

After 1 minute I can re-connect again to that PC with ssh just fine, the no-ip-client works great
BUT
I can not re-connect with my TS3-client.
If I stop the service of the teamspeak-server and start it again, I can STILL not connect.
I can see with top that the ts3server is running but no clients can connect.

I have to stop the service, start it by hand, with the start-script (clients can connect now)
stop it by hand, with the start-script
and then start the service teamspeakserver to make it run like before.

My question:

What is teamspeak-server keeping from noticing the new external IP?
Is it something to do with TS3server, with the network or is it due to systemd (that I do not know too much about).

There is no server.ini file used, just a straight and simple setup.

sshd works just fine, so in what way is TS3server different?

Thanks for reading and I hope someone knows what can be done.

Can not reconnect after external IP-change.