Whatever Gothmog was, he almost certainly wasn't going by his real name. The original Gothmog was Morgoth's Lord of Balrogs, who killed Fëanor and Fingon, before himself being killed by Ecthelion (a different one, not Boromir's grandfather
) in the Fall of Gondolin.
One question is, was Gothmog the "real" name of the Head Balrog? Tolkien doesn't seem to have decided what the origin of the name was. It may contain the same root as Morgoth, "the Black Enemy", which was actually the Sindarin translation of the name given to The Bad Guy Formerly Known As Melkor, by Fëanor. Was Gothmog the Balrog Lord's real name, or just the name by which the elves referred to him?
I can't see a human being given the name Gothmog at birth, although perhaps for an orc it's plausible. It seems strange, though, if Gothmog was really a Sindarin name of insult, why Sauron would pick that as a pseudonym for his lieutenant. It would make sense that Sauron would want to give one of his top lieutenants the same name as one of the top lieutenants of his former master (although Sauron's Gothmog wasn't nearly as fearsome), so maybe Gothmog was the Balrog Lord's real name.