[throwawaymath]

You're asking about two somewhat different things.

In the strict sense two things which are equivalent share the same properties, yes. This is the topological generalization of algebraic homomorphisms and analytic bijections. See the example about coffee mugs and donuts both being topological tori.

That being said I can't really comment on the potential artifacting details of this specific algorithm. In theory the overarching idea makes sense because if you find structure preserving maps between sets of varying dimensions you should expect relations within the set to be preserved (i.e. the relational information in the smaller set is equivalent, there's just less of it). But practically speaking not all datasets can be usefully abstracted to a manifold in this way, which means that (efficiently) finding an equivalent lower dimensional projection for the embedding might involve a fair amount of approximation.

With enough approximation you'll introduce spurious artifacts. But that's precisely where all the innovation comes in - finding ways to efficiently find equivalent structures with the representative data in fewer dimensions. This isn't the first proposal for topological dimension reduction (see information geometry); the devil is really in the details here.