I think all three problems are really one problem under the hood:
Are these two things actually the same thing, or they separate?
Are these two things actually the same thing, or they separate?
It's a great question, much deeper and more interesting than it seems. The essay suggests thinking in terms of isomorphisms (relative to the structure you care about) rather than equality in some absolute sense, and I've found a fuzzy version of that to be a really useful perspective even in areas that can't be fully formalized.
Years ago, I stumbled upon the "idea" was already debated in other fields long before programming. Lumpers and Splitters.