Is it possible to copy transactions between Bitcoin's forks blockchain? Do Bitcoin forks share the same transaction protocol after the fork? If so, is it possible to copy/ duplicate a transaction that uses common input transactions in the other blockchain?

Let me explain what I mean:

Let's say, I have 1 BTC in an unspent transaction which happened before a fork (Bitcoin Cash for example). 
So after the fork, I own 1 BTC in Bitcoin blockchain and 1 BCH in Bitcoin Cash blockchain.
Then, I send that 1 BTC to another person.
At this point, can that person copy my transaction code and re-send it in Bitcoin Cash network so that he gets 1 BTC and 1 BCH?
Yes, some people call this a 'Replay Attack'. But it's not literally an attack, but the desired outcome. If a fork don't want this to happen it can change the rules to prevent it. 

You can also avoid it but using inputs mined in BTC after the fork, if your wallet app allows this kind of selection.