Isn't the red Christmas outfit something Coca Cola came up with? So only since the 20th century? I think I've read the outfit was mostly green before that.
No, but it's a myth so common basically everyone believes it to be true.
The Santa we have today is an amalgamation of a few characters. The red outfit comes from Saint Nicolas. You can just look up "Saint Nicolas" and you'll find many fairly old depictions of a Christian bishop-like character in red garments carrying a bag of gifts.
The Dutch brought Sinterklaas (Saint Nicholas) to New Amsterdam, current day NYC, which has developed into Santa Claus as we know it know. Santa Claus is a phonetic derivation of Sinterklaas.
Apparently he was depicted as a thick-bellied Dutch sailor with a pipe and a green outfit in Washington Irving’s History of New York book, so maybe that’s where you got that idea from.