I don’t suppose that the Hamilton costume designer was going for strict historical accuracy. But the precise answer to the OP is that it is probably supposed to be George III’s Parliament Robe. That was worn by him during the opening stages of the coronation ceremony (but not for the actual crowning) and later whenever he addressed Parliament. It still exists.
It is not to be confused with the similar Imperial Robe, which he only wore during the final stages of the coronation ceremony (but, again, not for the actual crowning). That was purple rather than crimson. This is what he wears in the Allan Ramsay state portrait.
Incidentally, the Hamilton crown is certainly wrong. That is clearly based on the St Edward’s Crown, but George III never used that, instead only ever wearing an altered version of his great-grandfather’s state crown.
British peers also have robes with ‘ermine’, as do some Oxbridge academic gowns. But this is now usually only imitation, being either miniver or artificial. It has been indicated that the Queen will in future only use artificial fur on new outfits, but this does not affect her Parliament Robes as, being 94, she’s given up wearing bulky ceremonial robes.