Throughout the Deadpool movie after Ajax and Wade Wilson meet, Ajax keeps asking this,
What's my name?
Throughout the movie it seems to be a theme. Does it have some special significance in or out of universe?
This article explains this dynamic in one of its points:
6. What’s My Name?
Ajax and Deadpool have quite a history in the comic series. Ajax loves subjecting Deadpool to unfathomable torture while Deadpool loves messing with the fact that Ajax’s real name is Francis. Their relationship is perfectly translated on screen as Ajax asks Deadpool what his name is throughout the film. He wants to be called Ajax, but the merc keeps calling him Francis, refusing to bow down to his enemy’s demands. This running theme culminates into a scene that’s almost too awesome to be true.
Just before his final battle with Ajax, Deadpool successfully takes out a swarm of his henchmen armed with nothing but two katanas. After asking what his name is constantly throughout the story, Ajax finds Deadpool’s final answer literally laid out in front of him. The dead henchmen are revealed to be precisely arranged on the ground by Deadpool so that their corpses spell out “FRANCIS.” That’s a hell of a way to own your arch enemy if you ask us.
Ajax confidently tells Deadpool that nobody's sense of humour survives Ajax's "clinic". And the most manifest sign of Deadpool's smart-assery is his making jokes about what Ajax's real name is.
So, when - instead of breaking under the torture - Deadpool triumphs on that topic by finding out Ajax's real name (name tag "Francis" left over from dry-cleaning), it makes Ajax uber-concentrated to reverse this. To him, having Deadpool admit that his name is Ajax is a signal that he won and broke Deadpool.
(the idea of using "tell me the lie is true" as a yardstick of breaking your opponent is, if course, nowhere near new, and not only borrowed from the Deadpool comics, but from much earlier archetypes, such as "How many fingers" in Orwell's 1984).