JQuery equivalent of MooTools bind(this)

TBH, the mootools .bind as you call it is just Function.bind in ES5 - and is available natively in browsers that support the js 1.8.5 + spec. MooTools just enhances browsers that don't have it yet but lets the native implementation remain on the prototype - if available.

https://developer.mozilla.org/en/JavaScript/Reference/Global_Objects/Function/bind

You can easily implement that as a a Function.prototype.bind decorator if not available natively and use it as the example above says:

// Function.prototype.bind polyfill
if ( !Function.prototype.bind ) {

  Function.prototype.bind = function( obj ) {
    if(typeof this !== 'function') // closest thing possible to the ECMAScript 5 internal IsCallable function
      throw new TypeError('Function.prototype.bind - what is trying to be bound is not callable');

    var slice = [].slice,
        args = slice.call(arguments, 1), 
        self = this, 
        nop = function () {}, 
        bound = function () {
          return self.apply( this instanceof nop ? this : ( obj || {} ), 
                              args.concat( slice.call(arguments) ) );    
        };

    bound.prototype = this.prototype;

    return bound;
  };
}

As you can see, it's a little more involved than a simple .apply / .call

One thing to consider is, if you NEED to use bind or if you can save a reference instead.

eg.

var self = this;
this.target.bind("click", function(e) {
    var tip = self.opts.tip;
});

this has a smaller footprint than the function binding anyway. it also affords you a correct reference to this as the trigger element (event.target === this). you will find this pattern far more often in mootools-core than the bind one - though bind is often needed when you want to assign events to class methods, eg:

this.element.addEvents({
    click: this.showTip.bind(this),
    mouseleave: this.hideTip.bind(this)
});

In this case, saving a reference won't work though you can rewrite it as

var self = this;
this.element.addEvents({
    click: function(e) {
        self.showTip(e);
    }
});

A jQuery particular implementation is proxy - http://api.jquery.com/jquery.proxy/