Would you guess that Janos Slynt was born with his surname (which would be unusual as his father was a lowborn butcher), or somehow acquired it or invented it later in life?

If it’s anything like medieval England, the surname is due to the fact that there’s a half-million people living in King’s Landing and you can’t do what you do in a small local village, which is to distinguish between two people with the same first name on the basis of you know who’s related to who (so it becomes “Tom, Bill’s cousin” or “Tom, Sarah’s nephew”). 

However, it could be that Slynt picked up the last name when he became Captain of the Iron Gate or Commander of the City Watch, both of which are important enough posts that he might be considered worthy of a surname.