Stocks would continue to be denominated in shares. I've never heard of a stock being denominated in currency, but maybe there is one out there like that. A share is usually something like I own 1/N of the interest in the company so the formula of ownership is m/N where m is number shares you own and N is the number of total shares (some simplification here, but roughly accurate). Market clearing determines how much people will pay for a stock, which happens in all sorts of currencies.
Indeed, as I've said in the past, money is a technology, and different technologies can be applied to different purposes. The "major" world currencies are designed to be used as a temporary store of value, medium of exchange, and instrument of economic policy. Bitcoin could be applied towards different purposes.
>>> There's a lot of value in this though, no?
Sure, a subsidy is by definition a source of value.