JSFiddle - React, Tailwind, and code Playground

JavaScript

// Constructor
function Person(name){
    // You don't need to reference "self" here. It's already implied.
    this.name = name;
}

// Prototype
Person.prototype = {
    getName : function(){
        // 'this' in this case refers to an instance of Person. So, when creating
        // John, this.name will be John. 
        return this.name;
    },

    openConnection : function(host, port) {
        // Similar to getName(...), this refers to an instance of Person.
        // In your example, this.pointCount is NOT shared between John and Adam.
        this.pointCount = 0;
        this.ws = new WebSocket("ws://" + host + (port ? ':' + port : ''));

        // In WebSocket.onopen below, you're working with a new scope, so you won't
        // have access to 'this' as the Person anymore. You need to save 'this' 
        // somewhere, so you can reference it in the new scope.
        // *****
        var self = this;   
        
        this.ws.onopen = function() {
            // In this function, a new scope has been created. 'this' no longer
            // refers to John/Adam (The instance of Person), but to WebSocket
            // instead.
            
            console.log(this); // 'this' references the WebSocket instance
            console.log(self); // 'self' references the 'self' in the outer scope
                               // See *****
            
            // Since this = WebSocket in this scope, all we need to do
            // is this.send(...). If you'd like to obtain the refer
            // to the instance of the Person you worked with, you can
            // use the 'self' variable
            this.send(self.name); 
        };
    }
}

// Create two new person
var p1 = new Person("Jonh");
var p2 = new Person("Adam");

console.log(p1.getName()); // Prints Adam
console.log(p2.getName()); // Prints Adam

var TARGET_URL = "echo.websocket.org", TARGET_PORT = 7000;
            
p1.openConnection(TARGET_URL , TARGET_PORT ) // opens connection...