It sounds like, maybe, a core tiddler is messed-up or something is overriding a core tiddler ? Regardless, likely way over my head.
Just to clarify, which "part" of adding tags is broken?
The field in which you type the label for the new tag? The "+" symbol to create a new tag with the typed in label ?
In the sidebar's "More" / "Tags", can you access the Tags manager and does it work okay ?
What recently added plugins did you install? (Maybe un-installing one or more of those plugins might be required.)