Finder asks for password - moving files and tags

Only if the files on Macintosh HD are part of you Home Folder ( User name Folder ) Otherwise it can not an altered or modified.

The Operating System of Monterey and Big Sur both reside in a Sealed and Read Only Volume and changes can only be effected by Apple

I had a similar problem fixed by resetting permissions within the Get Info panel of the drive. In some cases I set permissions to ignore permissions. That worked too.