There's been quite a lot of interest in classifying fusion categories of (low) dimensions, and by reconstruction this effectively classifies the represented objects up to gauge equivalence. A few Arxiv preprints on this:

You can search the Arxiv for results specifically about classifications of certain Hopf algebras, of which there are many papers.

Gauge equivalence of $H$ and $K$ means there is a "twist"—a 2-cycle-esque object that modifies the comultiplication, but for which there is generally no structured homology theory—$F$ of $K$ such that $H\cong K^F$. The twists of the so-called "big quantum groups" are fully classified to my understanding (based on hearsay; I don't think it's been published), and the twists of the small quantum groups $u_q(\mathfrak{g})$ (which remain Hopf algebras, at least) are conjecturally classified by certain Belavin-Drinfeld triples on the Dynkin diagram of $\mathfrak{g}$ and certain twisted automorphisms.

There are also certain conjectures (well, they were conjectures a few years ago, anyway; possibly I'm out of the loop, but I think these were fairly deep and unlikely to have been resolved) in the theory of vertex operator algebras which would imply the existence of braided equivalences between the representation categories of certain twisted group doubles. Extensive work has been done to (successfully) classify such equivalences, mostly by Naidu and/or Nikshych.

So, yes, in short, a lot of people are interested in your latter question, though I only know of two cases where the pointed property is relevant:

- The small quantum groups are pointed, and these are of immense and active research interest on a wide range of topics for a wide range of reasons and applications. As linked above, a very recent paper of Negron concerns the determination of the gauge equivalence class of small quantum groups.
- A semisimple pointed Hopf algebra has the property that all simple objects of $\text{Rep}(H)$ are invertible. And the classification of pointed fusion categories is basically known (see the first set of links), and the most commonly encountered ones (in my experience, at least) are the $G$-graded (f.d.) vector spaces with associativity $\omega$: $\text{Vec}_G^\omega$. So the non-semisimple case would be more interesting; or at least more difficult.

It seems the pointed property might be more ubiquitous in combinatorial Hopf algebras, but these are outside my expertise. Otherwise, the quasitriangular case (rather than the pointed one) has been of substantial interest because braided fusion categories appear throughout a vast range of mathematical physics.