Have any actual developers looked at this? Speaking on something out of your depth saying it's impossible because you don't see how it's possible actually does more harm than good.
I write Javascript for a living and specifically use Websockets in production (I did a quick view-source and it looks like Chess.com is using Angular and probably a socket wrapper like Socket.io), so I see more possibilities here. If the server isn't smart enough about filtering user input then one client could send excessive requests to the other (a sort of peer-to-peer DoS), or, it could send malformed responses and cause an exception in the client-side code, causing a disconnection. Back in the days of AOL Instant Messenger there were programs called "punters" that did exactly this. Each generation makes the same mistakes as the last
For example, if a socket is shipping JSON-encoded responses back and forth, then improper parsing of input can inject invalid character sequences into the stream which cause JSON.parse() to fail (as the JSON standard demands that certain special characters called control characters be escaped in a \uXXXX format). That's merely one example (from a real bug in a real application). There's all sorts of possibilities, so don't write it off so quickly.
Sorry, but the only way that's possible is if they hacked Chess.com and either: