Just give me an old paint???

So WTH does this line mean? Obviously I've never head the expression (if there is one). Is this a type of horse with blotchy markings (Obviously I know nothing about horses), a racial slur or a reference to one who likes old paintings or likes to paint?

Washington's somewhat distasteful laugh after the comment suggested to me that it was a stab at him or put down by Hackman. Or maybe Hackman was just belittling the Arabian??? Then to follow it up with the Ulysses comment, which I didn't quite get either. The only two Ulysses I know of is our former president / commander of Tennessee Army and the Greek hero, which I remember reading all about back in the day and it was a bit shakespearean but not by any means hard to read or understand as I recall, although maybe that is what is meant by that line...

In any case I certainly don't get what the "old paint" refers to so if someone could shed some light on this I'd be grateful :)