What kind of server connects a remote system through the internet to local serial ports using tcp/ip?

A proxy server. A system requests information from the internet, and that traffic is routed through the proxy server before arriving to the requesting system.

This is why proxy servers are used for hiding identity and being more secure, as internet traffic would show the IP address of the proxy server, and not the requesting system, though this often depends on the type of proxy server, as some proxy server's main objective isn't anonymousity, but rather efficiency.