> It was his family name some time ago, so is this just the other sides' "Barack Hussein Obama" or is there more to it?
No, it's not the same thing. The Hussein thing is that the Alt-Right uses it to imply Obama being a Muslim (which he is not) and thus, by extension, a terrorist supporter/enabler.
The Drumpf thing, however, is nothing more than highlighting the family origin (which Trump actually even stated himself, per https://www.snopes.com/fact-check/donald-drumpf/).