User blog comment:Miskos3/Chapter 56, Review/@comment-36.72.67.45-20140415152131/@comment-7418318-20140416171206

Or he could share a relation with Mikasa's father, without the Levi and Mikasa ever meeting before they both ended up in the same organization, which is far more likely, since they didn't show any kind of recognition to each other.

Or they could even just happen to have the same name by chance. Since Ackerman is far from a rare name in reality, that might also carry over into the Manga and Isayama just gave them similar names to mess with us a bit.

Besides, we don't even know if Levi is that Captain Ackermans biological son, adoptive son, nevew, younger brother or anything at all. The only thing we know for sure is that Captain Ackerman was Levi's mentor or trainer.