As it turns out, I found this bug: Hyperion: Can't add second instance of Hyperion integration (multiple hyperion servers) · Issue #44684 · home-assistant/core · GitHub where this particular comment by the integration author pointed me towards identifying the root cause: Hyperion: Can't add second instance of Hyperion integration (multiple hyperion servers) · Issue #44684 · home-assistant/core · GitHub
I had upgraded one of my raspberries and simply moved the sd card from the old one to the new one. The internal id used by hyperion to identify the instance was not changed. When I installed a new instance on the previous raspberry, it seems that the instance id generated was exactly the previous one, so in the end I had two servers with the same id.
This is why homeassistant was giving me the message saying that the service was already configured.
To fix it, I could not find any way to change the instance id through the interface, so I accessed it through ssh, downloaded the configurations database (/.hyperion/db/hyperion.db) to my pc, modified it my hand and uploaded it back.
After a reboot I can now add the new instance to my existing home assistant setup!