How many IP addresses does a Com-Server having more than one serial port require?

Only one! Which of the serial ports is accessed is determined by which TCP port number is used.

Although in practice one always speaks of “the TCP/IP protocol”, we are really talking about two independent network protocols. IP is used to uniquely identify the network station while also providing routing mechanisms for cross-network communication. TCP is then layered over this protocol to use the addressing of port numbers for talking to an application running on the network device. This functionality becomes clear for example when you have a computer which is providing both an HTTP as well as an FTP server service at the same time. Both services are accessed through the same address, and differ only in their TCP port number (HTTP = 80, FTP = 21 etc.).

Additional information:
Manual TCP/IP Ethernet and Web-IO

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