Ensure that you are actually installing “Plex Media Server”, and not just “Plex”.
After installing Plex Media Server, you still need to “claim” your new server into your plex.tv user account. Read here how to do it: New Claiming Requirement for Windows
No, it is not necessary to create a new plex.tv user account.
Nor is it necessary to uninstall the server software from the old machine.
The only thing you need to do, if you have more than one server, and are using a manual port forwarding for remote access, is to create a second port forwarding rule in your router’s settings. This one needs to use a different public (WAN) port number than the rule for the older Plex server.
