... perhaps the dark brotherood also suffered under the years without empire.
There's still an Empire, it's just not in great shape. Otherwise, there wouldn't be a civil war over secession.
Also, I don't think the Morag Tong would make much sense in Skyrim, being primarily an aspect of Dunmeri culture. Yes, I know the Dunmer are spread out through the other provinces since Morrowind took nine kinds of beating, but I can't imagine Skyrim's government allowing a legalised assassin guild simply because some surly elves told them they should.