This is one of the many things the Disney series gets wrong. If I were to list every one of them, I would rapidly exceed the character limit.

Yes, it is true that in informal settings Crockett sometimes wore a hunting shirt (which, to our eyes, is more like a jacket, worn with a shirt underneath it). Hunting shirts were sort of the Carhartt jackets of the late 18th and early 19th century south: durable, hard-wearing outdoor garments. The buckskin part has come down to us through legend, though I'm unsure of its veracity. Most hunting shirts were made of linen, which is (in my opinion) more comfortable, dries quicker, and can actually be washed when soiled. There are some recollections from those who knew him, written down decades after his death when legend mingled with memory, that he wore buckskin, so take it for what it's worth. There are other accounts from his time in Texas that he dressed in ordinary clothing. We'll probably never know. But he did own an embroidered buckskin waistcoat.

With that said, all accounts of Crockett's time in Washington indicate that he dressed in an utterly conventional mode. Take a look at the following image:

https://npg.si.edu/object/npg_L_NPG.1.88

This is a portrait of Crockett painted in 1834, and in it we see a man dressed in the manner of a perfectly ordinary man of the 1830s. He wears a black broadcloth frock coat, seemingly double-breasted. His waistcoat is silk or thin wool - I can't tell which from the portrait, but both would have been appropriate. He wears a white ruffled front shirt and has a black silk cravat knotted loosely about his neck. There is nothing rustic or even old-fashioned about his attire: he's wearing the business suit of his era. Other portraits, of which there are I think five, similarly show him in black broadcloth coats, cravats, waistcoats, and white shirts.