you asked about EthernetClient, not about websocket. I do not know the websocket specification.
The EthernetClient created by the EthernetServer has always the same 'sock' number as the server has. It can be only one for that server at time.
I have a Telnet server at port 2323 in my project. Until the connection is open for a telnet client, no other telnet client can connect to the server on port 2323.