Wasn't this the plot of Iron Man 3?

Take away the suit, and Stark is still the guy who made the suit.

Not at all.

He's a billionaire genius playboy philanthropist.

Stark made the suit with no help. He doesn't need a specific suit because he has the skill to invent whatever he may need.

At this point, Peter can't make himself a suit like that, so if he is nothing without it, he can't respect the power it brings. But he isn't nothing without it, which is what Stark is trying to teach him: not to rely on power of others.

Steve Rogers: Big man in a suit of armour. Take that off, what are you?

Tony Stark: Genius, billionaire, playboy, philanthropist.

Tony was being snarky, but he's not wrong about the suit being just an extension. Yes, it's important to his abilities, otherwise he wouldn't have needed it from the beginning in the cave. But it's also not a crutch, as Ironman 3 showed and taught him, and he's trying to show Peter that it begins with the person.

Also keep in mind what he said to Peter in this same scene - when Peter said he just wanted to be like Tony, Tony comes back and says yeah, and I wanted you to be better. Tony knows that Peter truly doesn't need the suit because he is that powerful, it's just once again an extension that enhances those abilities, and if Peter thought it was the suit that made him special he wouldn't grow.

The suit is merely a product of his true power