Are shares bound to a currency?

I doubt that you can (at least not through an exchange since the exchange only deals in one currency), but even if you could:

Why would you expect that the fair value of the stock in another currency would be any different that the equivalent value in another currency? Otherwise there would be an arbitrage possibility - buy the stock in USD, sell it in currency XYZ, and convert XYZ to USD.

In other words, the value of the stock in currency XYZ should be the same as the value in USD converted to XYZ.

Shares are NOT bound by a currency - PRICE QUOTES are. A share is a share is a share. YOu own a part of the company that corresponds to your number of shares vs the total number of shares. The price of the share does not enter anywhere in this relationship.

Now, shares are quoted on various places (one, but possbily more) and those may be in different countries, and those MAY be quoted in different currencies.

Is it possible for Person A to sell the shares in another currency than the person have bought it?

Generally no. A Price Quote is on an exchange. YOu trade vs the exchange, not another person. So, both parties must share the same currency / price feed.

Now, it is possible that the counterparty is a market maker that is arbitrading a price - if you buy in NY (USD) and sell in Germany Frankfurt (EUR) you MAY have a situation that changing the EUR back into USD may give you a SMALL profit. Then the share you buy may originally come from a US seller. Nice. Except - that are TWO TRANSACTIONS, with a middle man, so you STILL handle YOUR side of the operation within one currency.