"One of the two "slacks" I'm a member of recently got a UI update - and the other didn't."

This is because major changes at basically all business software companies are rolled out on a team-by-team basis, rather than per user account.

Also note that from the screenshots that he is on Free tier Slack teams in both instances. When you do not pay for a business communication tool, you should not really have expectations of stability or coherence. You're getting a free sample of a product that is meant to be paid for. What he was experiencing was a beta or gradual rollout where Free teams were being used as test subjects to see how the redesign impacts user engagement. If you don't pay for a product, you are the product.