My car is back online. 
tl/dr
I had to change the APN settings. I believe there is a bug in Android 10: If the client of the WiFi hotspot only supports IPv4, but the APN for the mobile network is configured for IPv4IPv6 then the mapping of the IP addresses seem to fail. Please give this information to your develop department.
Full story:
I noticed that the APN settings of my Telekom SIM card are different from those documented on Telekom’s website, so I decided to change the settings manually. The result was a disaster. I completely lost mobile data connection. Worse, after resetting the APN settings (using the built-in reset function) and rebooting the phone, mobile data still did not work at all. Great. 
I removed the SIM card and put it back into my old FP2. It worked like a charm. Then I noticed that the FP2 uses entirely different APN settings that neither match the default APN settings on FP3 nor the settings documented by Telekom. So I decided to write them down on paper, put the SIM card back in the FP3, and manually edit the APN settings on FP3 to match those on FP2. Finally some cache cleaning, a reboot, and - voila - mobile data was back again. 
And even better: now the car can access the internet again through the hotspot. Yipee!
One difference between the old and new APN settings is the protocol. The old (working) uses IPv4 while the new (not working) uses IPv4/IPv6. Here is my hypothesis:
- The Toyota only supports IPv4 (?)
UPDATE: my original guess seems to be wrong. I got me the adb-tools and checked the log files. It seems that the car uses an IPv6 address. Must do further investigations. - Connection to the mobile network requires IPv6 (suggested by the APN’s name internet.v6.telekom)
- So the phone has to map IP addresses when routing the traffic from the car to the network and vice versa. And that mapping does not seem to work properly.
I’m also relieved that I didn’t need a factory reset that probably would not have fixed the problem.