"Nothing" as the FritzBox is dead, or your connection to the HomeAssistant green disappears? While it is up, what is shown, MAC, IP Address, port number, Time to live, etc? Is the time it stays up consistent before it crashes?
Interested to know what the FritzBox CPU and memory usage is with your setup. Statistics on network errors and retries would also be valuable. You haven't mentioned which FritzBox model and firmware revision level you are using. This would be very useful information.
Have you involved FritzBox technical support in your troubleshooting path? Yes, I know you have had the device replaced, but are they still tracking the problem, and what have they suggested? No point in duplicating troubleshooting in both forums if the results are consistent. The issue is probably the correct question hssn't been asked yet, and the '"it still doesn't work" is a little too broad to work with.
Wireshark may offer insights as to network traffic.
Try it without the green HomeAssistant server connected first, and then with it switched on and compare the difference. Try it again with the Raspberry Pi version instead.
Alternate option to check: Is your power supply to each and every device reliable? Your home switchboard, (particularly the neutral and earth links - if installed - are functional and not sparking or loose)? Your UPS? Each plug pack/wall wart? The green? The Raspberry Pi? The FritzBox? Any power boards and extension leads? What is your actual measured home AC voltage, not the nominal voltage?
Use the process of elimination to narrow down the issue, leaving everything off and adding one thing at a time and observing what changes. After a few different combinations, a pattern should emerge that will point towards the underlying issue.
If the problem is widespread as hinted, your carefully documented process of elimination should enable the solution to be found. Yes, it may be the FritzBox, but you may have to narrow it down to whether it is hardware, firmware, drivers, or a rogue app that causes the problem in the first place.
As part of troubleshooting, Fritz technical support may offer you alternate customised firmware, hardware, etc to narrow down the problem. I'm certain they are following here (I hope you have pointed them to it). It would be nice if we could have a cross reference to their support forums here as well. It is in their very best interest to avoid ruining or boosting their reputation, something this thread is contributing to. The choice offered to just go with another router or HomeAssistant platform may or may not fix your problem, but how about everybody else following along - is it something unique to a particular combination of factors and how to break that chain? Don't leave us hanging!