Basics of a Node
1. The entire blockchain is stored on the node itself.
2. It consumes less power

Basics of a Browser
1. A browser is just a front end visualizer to the user.
2. It uses very less resources to store values in the form cookies.

A browser in my aspect can't be a node, as a plugin can't handle a blockchain. If the entire blockchain is stored in the computer where the browser is installed, then the qt itself can act as a node. Why do we need a browser to act as a node?

So, it is not possible to make browser a node. There is no possible for a light weight node in terms of memory usage.