The movies with the constant and incorrect "I Am Iron Man" have led people to think that Tony is Iron Man even without the armour.
The below quote explains it perfectly.
Tony Stark is who is underneath and Iron Man is what he can do. The two are not mutually exclusive, no one else can be Tony but others can be Iron Man.
Lets look at some other superheroes on film.
If you remove the costume from Captain America then Steve Rogers can still be Captain Americain in that he can do what Cap does as he still has the same super soldier abilities.
If you remove the costume from Batman then Bruce Wayne can still be Batman in that he can use his training & expertise to defeat the villains.
If you removie the costume from Daredevil then Matt Murdoch can still be Daredevil in that he can do what Daredevil does as he still has the same hightened abilities.
If you remove the costume from Superman then Clark Kent can still be Superman in that he retains the same abilities.
If you remove the costume from Iron Man then Tony Stark CANNOT still be Iron Man as he can't do what Iron Man does.
In Iron Man when he is without the armour he is easily incapacitated by Obadiah Stane and his friend has help save him.
In Iron Man 2 when without his armour he barely escapes alive from Whiplash when his friends help save him.
In Iron Man 3 he displays some resilience when storming the mansion with tech and guns, he is captured though. Has to be saved by his girlfriend when he fails to defeat the villain.
It's clear that without his armour Tony Stark isn't capable. Ne needs the suit to be Iron Man.
Thus Tony Stark is not Iron Man (when without the suit), he needs it to be a superhero.
Iron Man and Tony Stark are two seperate characters.
And even then it doesn't need to be Tony piloting Iron Man. Iron Man 3 showed that people other than Tony can pilot the suit so in effect Iron Man can be anyone... As long as Mr Stark is there to build the suit first, of course.