This appears to be a bug in Golden Gate. Bugs should be reported to Apple here:
Feedback - macOS - Apple
Note that it's also possible this is an intentional UI change, but that seems unlikely in this case given that the grouping functions remains a part of macOS. However, I cannot find documentation of the 'move the group by dragging the app' feature. Even if it's intentional, feedback may get Apple to change it back.
On macOS 27 Golden Gate, I find that Mission Control and Spaces behave mostly as it did on prior versions of macOS, and as expected based on Apple's information.
View open windows and spaces in Mission Control on Mac – Apple Support
Work in multiple spaces on Mac – Apple Support
However, there are a couple of behaviors that are different and this is one of them. I can replicate what you see. The Group windows by application setting must be toggled on.

On my Mac running Tahoe, with clicking the app icon in Mission Control in a Space with multiple windows of that app open allows me to drag the whole group together to another Space.

On my Mac running Golden Gate, clicking and dragging the app icon leaves the app icon behind (unlike Tahoe) and only drags the top window to allow it to be moved to another Space.

You can certainly try various troubleshooting steps, as follows:
See if the problem is present when you boot in safe mode. That process performs some system cleanup and clears caches, and also disables 3rd party extensions that may cause problems.
Start up your Mac in safe mode - Apple Support
If that does not correct the issue, test in a new user account. A newly-created user account will have all settings at default, and if the issue does not occur in that new account then it provides guidance on what may need to be changed to correct the issue.
Add a user or group on Mac - Apple Support
Once the testing is complete, the new account can easily be deleted.
Delete a user or group on Mac – Apple Support
However, given that I see different behavior on two Macs sitting side by side with one on Tahoe and the other on Golden Gate, this does appear to be a UI bug that Apple would need to fix.