Socket.io keep the connection on client side when client refresh or open new page

It's impossible. You cannot keep the same socket.io or webSocket client connection when the page is changed or refreshed. The browser simply does not do that. When a new page is loaded or the current page is refreshed, all resources from the previous page are closed and freed by the browser, including socket.io/webSocket connections.

So, your server has to expect a new socket.io connection from the newly loaded page. If you use cookies or a server-side session object, you can identify, on the server, when a connection is coming from a client that you have previously seen and the server can then act accordingly to realize that this is just a previous client reconnecting on a new page.