What is the stuff in Play Doh that gives it that smell?

Diced baby fingers.

Big Bird’s droppings?

“Play-Doh’s current manufacturer, Hasbro, reveals the compound is primarily a mixture of water, salt, and flour, while its 2004 United States patent indicates it is composed of water, a starch-based binder, a retrogradation inhibitor, salt, lubricant, surfactant, preservative, hardener, humectant, fragrance, and color.”

So, to answer the OP, the stuff in Play Doh that gives it that characteristic smell, is fragrance.

Right
So the question stands

Lots of safe things have characteristic smells due to added fragrance of undisclosed chemistry. Baby Shampoo, Ivory Soap,baby powder and baby oil, even fingerpaint.

Sometimes, producers of fragrance oils can replicate these fragrances (and lots of other unusual and characteristic ones)—there is no way to know how they acquire the chemical formulas as they are probably trade secrets.