Unable to save document in Preview after adding tags

Have you tried using File menu -> Save As PDF and directing to your Desktop, Documents, or Downloads folder?

Is iCloud syncing any of your folders? That could stop the read/write access to some of your folders. For grins go to your Hard Drive -> Users -> Shared folder and see if it saves there. If it does, then either Time Machine or a cloud syncing happening on your account is preventing file saving while it works.

The Preview Inspector panel will allow one to add keywords (not Finder tags) to images and PDF documents without any loss in the ability to save that opened content via a Save menu item or its associated keyboard shortcut.

Those keywords can be seen in a Finder Get Info, or located with Spotlight as in the keyword "Woof" placed on an image file:

kind:png keyword:woof