Matt didn't grow up in any kind of orphanage in the comics. Comics!Matt was raised by Jack all through his childhood, since comics!Jack didn't die until Matt was a young adult.

TV!Matt grew up in a Catholic orphanage. In episode seven, Stick says "Old bitty at the orphanage thinks it's her idea," in reference to the nun who found him to help Matt. And in episode ten, Matt says that Stick found him in an orphanage.