I haven’t used the technology directly, yet… but my understanding is that the negotiation of a websocket first happens over HTTP. Once the negotiation is complete, it becomes a websocket. Just as HTTP can be secured with SSL/TLS to make it HTTPS, WS when negotiated over HTTPS is WSS.
So a server would presumably use the exact same certificate since web traffic and websocket data both happen through the same negotiation. WSS would typically happen on port 443… but just with HTTPS, you can tell the server to use other ports and thus you could have two different servers on different ports, one for web traffic and one for websocket connections.
You’d need to explain what type of server you’re using to say much more on the subject.