The problem you’re describing doesn’t really click for me as for how you want the final result to look like, so maybe there is indeed some misunderstanding about how things work and are supposed to work.
If you only need to send messages between users, you may not need a server component at all. A quick Google search tells me that Socket.io has the concept of Rooms that lets you send messages to groups of players directly. If you could make a room for every game match, you could just send game moves directly from player to player without any additional logic on the server side.
But to be clear: I’m not that familiar with Websockets in general, and definitely not with Socket.io, so don’t take my word for it.
If you have a backend in Node.js, you cannot host it with us, because we don’t provide Node.js hosting.
In theory, you could run your Node.js backend on Repl.it, and transfer data to another server from there. I would normally suggest making HTTP calls to the website backend hosted on the web hosting, but this will not work here due to this security system: Ensuring only web browsers can access your website
But you could also ask yourself whether you need any data to be sent to our servers in the first place. If all you need to do is send game events between two browsers, what purpose do our servers have in this?