In said Arthurian romance game, however, I'd like to think nobility or a castle would be something a character could aspire to as an any-term-length goal; and maybe having (or gathering) lots of money might be one of numerous avenues toward achieving said goal.
It was a thing in the real world, but it's not really a thing in Arthurian romances. The Knights of the Round Table and knights aspiring to the Round Table don't engage in commerce nor do they collect wealth in the pursuit of their personal goals or in return for their good works. Though I grant that I might not be remembering the part of Malory where Agravain trades in the favors of noble maidens that he's won in jousts for a small keep in the Midlands.
So, where in the literature and even modern popular media of this genre is money a focus. I'll assume it manages to get mentioned in some modern source, Mallory never talks of money at all! In the context of the genre buying a title would be laughable, such a person would be like a talking pig. Nor does the sort of cash economy required seem to exist.
Just to echo the posters above, within the genre, a noble might bestow lands, a title and castle for great deeds and service faithfully rendered. Or a wicked knight might steal a castle and its lands, dispossessing its former occupant. Or it might be acquired as part of a quest, after the witch who lives there is slain - with the blessing of the local lord, of course.
