As you can see on the screenshot, Bear is still showing in stage manager (also on every space) while the app is in full screen mode.
Just adding my own observations. I do not see this behavior when I enter full screen mode on my Mac (Beta Bear and Ventura). Everything is working fine.
Adding my own observations as well. I see this exact same behavior, but it is not specific to Bear. Safari and Things will also do it when I have multiple windows in Full Screen mode.
I have not experienced this issue with Stage Manager but I suspect this is a macOS glitch. Maybe check the macOS 13.2 update.
After updating to 13.2 yesterday, I was not able to replicate the issue. Agree it was a MacOS glitch
Interesting. For me it is still here on 13.2. Anyway, I will send a feedback to Apple. Thanks!