If you want to tinker with an engine as oppose to making a Web game, I would suggest Phaser too.
Then there is QICI Engine that seems like it's no longer maintained, but it's source code is still on GitHub. It's built on top of Phaser. It is a full integrated web-based game engine as oppose to Phaser which is a framework, also Unity-like, since it has scenes / gameobjects / components etc.
If you're doing something HTML5, try QICI Engine. It's released under the MIT License. It's a HTML5 game engine that runs in a browser. It has a Unity-esque interface, but underneath it's all Phaser. EDIT: Oops, it's 2D only.