When I drag a tag on the questions page from the tag list to the input field Favorite Tags it breaks:
The tag usability is changed to
It's not designed to, I'm afraid. See this post on Meta.StackOverflow and the associated response:
There is no drag-and-drop functionality with the tags right now.
What you did, and can do on most browsers, is just grab the URL of the thing and it will perform a paste right into whatever input you're going for.
You can also do what you did with avatars or links. But it doesn't mean it's drag-and-drop.