I really like this game idea. It's a cool take on two-player snake with a tug-of-war twist.
One thing I found a bit tricky was that pressing the up/down arrow keys scrolls the web page wrapping it. Would it be possible to update it with a call to Phaser.Keyboard.addKeyCapture in Doggo.Game.prototype.create? That would prevent the page from scrolling, I think.
Otherwise, this is so cool! Great job!