Jellyfin is an open source fork of Emby.
The Emby integration works fine, but this is not obvious/findable, as it doesn’t mention Jellyfin anywhere. (The Jellyfin site, understandably, also focuses on Jellyfin features instead of on being an Emby fork)
As long as it works with both, duplicating the code doesn’t seem like a great solution. But maybe there is some way that searching for Jellyfin could find the Emby integraton, and the description could mention it also works with Jellyfin? (Or even rename it “Emby/Jellyfin”)
Personally, I found Jellyfin while looking for an open-source Plex alternative and it took me quite a while to discover that it was possible after all to integrate it into my Home Assistant setup. Which made both projects seem slightly less awesome than they are.