If you have a limit on the number of modbus clients then dropping old connections when a new one arrives is actually suggested in the modbus implementation guide (https://www.modbus.org/docs/Modbus_Messaging_Implementation_Guide_V1_0b.pdf)
Nevertheless a mechanism must be implemented in case of exceeding the number of
authorized connection. In such a case we recommend to close the oldest unused
connection.
It has its own problems but everything is a compromise.
Regarding supporting multiple clients…if you think about modbus/rs server – it could only ever have one master at a time. Then replace the serial cable with TCP and you see why it’s not uncommon to only support one client (and of course it’s easier to program). It is annoying though.
Depending on what you are doing you wont need the whole modbus protocol and implementing the parts you do need is pretty easy. Of course if you have to support absolutely everything its a different prospect. I haven’t used freemodbus, or any other library appropriate to your setup, so I can’t help with suggestions there.
Regarding the PC using different TCP source port each time – that is how TCP is supposed to work and no fault on your side. If it did reuse the same source port then it wouldn’t help you because e.g. sequence numbers would be wrong.
Regarding dropping clients. You are allowed to drop clients though its better not to. Some clients will send a modbus command, notice the connection has failed, reconnect, but not reissue the command. That may be their problem but still nicer to not see it that often where possible. Of course things like battery life might make the calculation different.