Assuming you are asking "what is practical & expected", not what is possible.
A Modbus/TCP slave limited to only 1 client is pretty unusable. The most common situations are:
1) customer has 2 clients (master & backup) plus maybe a field tool, so needs at least 3 sockets.
2) during network hiccups, a client might go away and need to reconnect BEFORE your server frees up the socket, so a few sockets need to be available.
If I were creating such a tool, I'd offer AT LEAST 4 sockets, but 8, 10 or even 16 is safer. However, I think if you offer 4, no one can throw MUD at you! But don't go less than 4. 255 is crazy (& why stop at 255? TCP ports are 16-bit, so you could support 10 or 20 thousand clients! :-] )
As for serial, that is always 1 master/client per RTU serial port.