The caffeine version in F44 is incompatible with the gnome-shell version in F44. This means the package does not work at all. It should either be updated or removed from F44. Reproducible: Always
This is fixed in version 60-1.fc45 (working well on Fedora 44 in my testing) Version 60 needs to be backported to 44.
Can confirm the same issue here, happened when upgrading from F43 to F44 and also reproduced it on a freshly installed Fedora 44 machine. Installing the package using dnf works but the installed version shows up as/is incompatible with Gnome 50.