Yes, you can say both in Danish as well. Both are equally good. "Hun tager sin frakke af" and "Hun tager frakken af". BUT, the word FRAKKEN, is actually not the most used word, the more broader description is JAKKE.
I think BaconChomper meant does 'tager af' always have to be separated by the object in the midde, or can it be together with the object after? e.g. 'hun tager af sin frakke'? In English it can be both ways, I'm also curious if this can happen in Danish too.