The domain company that you talked to may have done a poor job of explaining how domains work. Domain names don’t refer to specific ports. They just refer to IP addresses. The client can look up a hostname to get the IP address which the client should connect to, but the client has to figure out the port without the help of DNS. Port 80 is just the default port for HTTP service.
You can certainly run a web server on port 8088 if you like. The port number would have to appear in the URL, e.g. http://somehost.example.com:8080/some/page. Clients would parse this and know to connect to port 8080 instead of the default port 80.
If you don’t want URLs to contain the port number, then requests are going to go to the default port 80, and you have no choice but to make the web server running on port 80 handle these requests. HTTP/1.1 requests include the hostname which the client wants to contact, and modern web server programs are normally capable of serving completely different sets of content based on the hostname in the request. There are few ways todo what you need:
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Just configure the web server for port 80 to handle both sites. This will depend on what web server software you’re using. Apache for example calls these “virtual hosts”, and here is a set of examples. This is a typical solution, and some people run hundreds of sites on the same server this way.
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Run your two web servers as you planned. Set up the server for port 80 to be a reverse proxy for the second website. The server would continue to serve content for the site it handles now. When it receives a request for the second site, it would relay the request to the server running on port 8088, and relay the server’s response back to the client.
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Move the existing server for port 80 to a different port. Run a pure reverse proxy server on port 80, relaying requests for both web sites to their respective web servers.
You might be better off taking further questions to https://webmasters.stackexchange.com/ or https://serverfault.com/.