It’s called a patronymic, and is also part of the legal name in At least Ukraine and Russia . Interestingly in those countries the last name is something which only really shows up in official documents, instead the patronymic is used in much the same way as a last name is in English. For instance the Russian equivalent of John Doe is Ivan Ivanovich which just means Ivan son of Ivan.

I've made a mistake of using my patronymic as my middle name in US docs. It's actually more like a second last name, as you noted.