Kagawa definitely was a dribbler at Dortmund. For whatever reason we havent really seen much of it from him at United so far
I think it's a combination of 3 things:
1. With more mobile teamplayers(Dortmund) around you, it's easier to dribble because the opponent has to keep track of more people. A sudden burst might therefore trick 2-3 players.
2. Confidence. He's new to the PL, and he even though he's had a pretty decent start, he says himself that he's not satisfied with the way he's played(which is understandable, considering how well he's played for Dortmund and Japan).
3. While the Bundesliga isn't that far behind the PL in terms of general football level, there is one thing the PL can boast: the toughest set of defenders in the world. The tempo is also a split second faster.
Combine these three reasons, and it's easy to see why it will take Kagawa at least a few more games before he can reach his former glory. He has a split second less to think, less mobile teamplayers, and face the toughest defenses out there. This has ultimately given his confidence a knock. It's tough to go from being one of the best players in the Bundesliga, Asian player of the year, and being named "World Class" in several news papers, only to become a "decent United player". It's quite a drop, if you ask me.
But the guy is a fighter, and he's only 23 years old. It's way too early for him to stop developing.