In my opinion many of the same emotional response research techniques apply.
What UX and CX have in common is that both are a result, a consequence of how a strategy fulfills expectations set by a product.
User Experience (UX) refers to a person's emotions and attitudes about using a particular product, system or service.
A brand is in essence a promise to its customers of they can expect from their products, as well as emotional benefits
Another Wikipedia Definition
So to address your tactical question:
How do I measure the brand perception and how do I compare the current brand design with a redesign (validated by users)?
There are many ways to gauge emotional response to branding. Product reaction cards and memory tests are some of my favorites. Show people style tiles or mood boards and see if different variants elicit different surveys ratings, or different adjectives.
As far as the pairwise comparisons, yes you could snap a baseline with your current branding and then compare audience reactions to your intended brand attributes:
- "We're going for 'sh*tty punk.'"
- "We want people to think 'They're like Uber, but for bikers.'"
Then you can redesign whatever aspects of it and just repeat the experiment. If you design clean tests and know what brand attributes you're aiming at, you'll probably get some answers out of that work.