Shallow equality checks do only compare references, which makes them rather cheap.
const x = {};
const y = {};
console.log(x == y);
// prints `false`
This fact also does not change if you use React. However, your twitter post is very misleading. They don't actually mean shallow JS object copies, but an algorithm used by React's PureComponent in one of their methods (see docs):
React.PureComponent’sshouldComponentUpdate()only shallowly compares the objects.
I don't know a lot about React, however I think that this means that they probably have an algorithm, which compares first-level children for equality. I don't know how many elements that sums up to, however it's a loop in JS, which makes it slow - and if you do it for many components, you add up costs in a blocking way.