ES6 class that generates an area where you can draw on a page
The first weird thing I saw in your code is the use of the $(fun).

This jQuery function is used to execute code when the DOM is ready.

I think it should be better to remove all this calls and instead instantiate the class itself in a function executed after the DOM is ready.

That helps to follow the code properly.

// here you declare the class
class MyClassThatUseDOM {
    constructor(selector) {
        this.myElement = $(selector);
    }
}

$(function(){

    // here you instantiate the class 
    // when the DOM is ready and all selector
    // will properly match.
    const myObj = new MyClassThatUseDOM("#theselector");

});


As the element $(selector) or $(this.selector) are used may times it is best to save in an instance variable:

this.element = $(selector);


And then access always from this variable, instead of continue to search it with jQuery.

The following method have some issue for me:

createChessBoard() {
    var id = this.selector;
    $(function() {
        $(id + " tr:nth-child(even) td:nth-child(even)").addClass("black-tile");
        $(id + " tr:nth-child(odd) td:nth-child(odd)").addClass("grey-tile");
    });
}


Maybe if you remove the $(func) part as I suggested in my first point the code will be more clear.

But let's discuss what we have now.

You use a variable id to save the selector, but id is more specific term than selector. It could confuse some developer.

A selector for jQuery is anything is a valid css selector, like a tag name, a css class name or even and id.

Please pay attention to the names you use for your variables.

In this case you can do something like:

let _selector = this.selector


As you can see you understand that you just assign a private variable, here.

Use of var is not wrong, but we have let.

Similar considerations for:

var bname = $(this.selector).css("border");


About the method:

createRandowDraw() {
    this.clearBoard();
    let mat = $(this.selector + " td");
    let max = mat.length;
    for (let i = 0; i &lt; max; i++) {
        let num = Math.floor(Math.random(2) * 2);
        if (num &amp;&amp; num &lt;= 1)
            $(mat[i]).addClass("tile-active");
    }
}


Here you could use jQuery API to simplify your code:

createRandowDraw() {
    this.clearBoard();
    $(this.selector + " td").each(function(i, mat){
        let num = Math.floor(Math.random(2) * 2);
        if (num &amp;&amp; num &lt;= 1) {
           $(mat).addClass("tile-active");
        }
    });
}


Or in a plain javascript fashion by using filter and forEach.