The Name’s Castiglione
After the opening of the first episode establishes that he’s hung up his guns after killing the last of the gangs who murdered his family, we join Frank Castle as he tries to keep his head down and lead a quiet life. As he’s a wanted felon and meant to be dead, Frank is living under an alias: Peter Castiglione.
Punisher fans will immediately pick up on the significance of this surname. In the comics, Frank was born Francis Castiglione. Under this name, he became a highly-decorated U.S. Marine for his fearless efforts during the Vietnam War. After his second tour of duty was up, he changed his name to Frank Castle so that he could illegally set out on a third.
For simplicity’s sakes, this storyline is not addressed in The Punisher, but it’s potentially still part of the character’s backstory. That would explain where he gets his cover name from. As for the Pete part, Google tells us that Pete Castiglione was a Major League Baseball player in the 1940s and 50s. So maybe Frank’s a fan?