in simple says the text: bms has no problems with multiple computers in a network when nat in the router is on. this can be checked if you can access websites with both computers at the same time.
the problem is apparently a broken or changed configuration of the game and therefore you have to proceed as semler has written.
that is not completly correct. there are many (game)servers that have exactly this problem but there are also many who ask the router for internal ips and ports or can be guided on. but the server has to be able to do that and your router too, otherwise you are right.
To keep the computers apart on the server there are several possible ways. there can be used almost anything that is specific. gamekey, hashes, mac-adresses, etc ... but for everything that comes back from the server must be determined which computer to be addressed.
whether bms or pr I do not know but I think you can believe what semler said.
quake3 for example i already played with 2 computers over the same network in the internet.
another possibility in which it would have to go in any case is, if you connect one of the two computers via VPN with another network and then from there to the internet. then you have two different ips outwards although the computers are attached to the same router.