So, Bitcoin Cash and Bitcoin both use the same address system?

I personally got into Bitcoin long after Segwit, so I have no BCH.

But let's say someone owns both BTC and BCH after the hard fork.

And then, he wants to sell BCH and buy BTC.

However, when he broadcasts a transaction on BCH, what is to stop someone from broadcasting the same transaction on the BTC network?

Worse, since Segwit essentially creates a "anyone can spend" UTXO as interpreted by pre-Segwit nodes, if he transfers BTC into a Segwit address, what is to prevent someone from stealing his coins on BCH?