Apps still open on the old machine wouldn’t affect this machine.
There is a file that is used to tell if you are currently signed in. It may be that file is in a confused state.
In Finder, choose Go to Folder…, and paste in this path:
~/Library/Application Support/SketchUp 2025/
Delete the file named login_session.dat. Do the same for older versions of SketchUp, like this is the path for 2024:
~/Library/Application Support/SketchUp 2024/
Once those are gone try reopening 2025.