I've bolded the parts where the player is superior. Italics where they are equal.
Carrick:
Passing accuracy: 88.1%
Successful through balls per game: 0.1
Key passes per game: 1
Successful crosses per game: 0.2
Successful long balls per game: 5.8
Assists: 4 assists in 3044 minutes
Kagawa:
Passing accuracy: 89.7%
Successful through balls per game: 0.3
Key passes per game: 1
Successful crosses per game: 0.3
Successful long balls per game: 1.5
Assists: 4 assists in 1314 minutes
Carrick only beats Kagawa with long balls, but that is only to be expected, given his position on the field. It's also worth to keep in mind that Kagawa is in a more advanced position, which makes it harder for him to pick out the right pass and gives him less time to think. Carrick is awesome, but a lot of his strength comes from his intelligence, positioning, and defensive efforts. When it comes to passing in general, Kagawa slightly edges it. I know that stats aren't everything, but they're not completely useless either.
Either way, passing is not even considered one of Kagawa's biggest strengths.