I have zero information on how X (i.e. Musk's holding company for Twitter) is structured but it's possible to issue stock in a private company. The difference is that you cannot sell that stock on the public markets and need to find a buyer through a secondary market. Your requirements for selling are also typically more restricted.

Restricted in like you cannot do it unless the company oks it.

Hopefully not that but it could happen. Typically you’ll have a lock up or the company has a right of first refusal to repurchase the stock.