As far as I know, grounding in the sense of set-theoretic geology isn't intended to provide a formalization of talk of metaphysical grounding, and I doubt it was even intended to provide a case of grounding in the metaphysical sense, although it's not a terrible stretch to think of it as such a case.
Remember that Paul Cohen proved that the negation of the Continuum Hypothesis (CH) is consistent with Zermelo-Fraenkel set theory with choice (ZFC), as follows. Given a model M for ZFC, and given a particular kind of sentence S about set theory, Cohen explains how to construct a model of ZFC which is larger than M in which the sentence S is true. That larger model is called a "forcing extension of M." It requires some care to specify what kinds of sentences S are allowed, but Cohen shows that it works if S is the negation of CH. So if ZFC has a model, then there is a larger model of ZFC in which CH is false. So ZFC cannot prove that CH is true.
Hamkins and Loewe, in their article "The Modal Logic of Forcing" https://arxiv.org/abs/math/0509616 , talk about forcing extensions of ZFC in a way that philosophers will find very familiar. Start with a model M for ZFC, and consider forcing extensions of ZFC as possible worlds, in the sense of the possible worlds semantics for modal logic. We should have a Kripke frame of possible worlds, and we do, as follows: M' is accessible from M iff M' is a forcing extension of M. So a modal sentence "P is necessarily true" becomes "P is true in every forcing extension", and a modal sentence "P is possibly true" becomes "P is true in some forcing extension." Some ideas in the theory of forcing have nice clean formulations from this point of view, which you can see in the Hamkins--Loewe paper.
In the Fuchs--Hamkins--Reitz paper that you linked, they are thinking about the opposite kind of relation between worlds: rather than start with a model M for ZFC, and ask about the "possible worlds accessible from M" (i.e., forcing extensions of M), in the Fuchs--Hamkins-Reitz paper the idea is to start with a model M for ZFC, and ask about the possible worlds from which M is accessible, i.e., (transitive class) models of ZFC of which M is a forcing extension. It's the reverse of the modal accessibility relation that Hamkins--Loewe describe.
If you start with the model M for ZFC, and take the intersection of all the (transitive class) models for ZFC from which M is accessible, the resulting model for ZFC is called a ground of M. The idea is that, rather than going up into the stratosphere of more set-theoretically exotic collections by taking larger and larger forcing extensions (as you do when considering set-theoretic "possible worlds"), when you speak of set-theoretic "grounds," you are going in the opposite direction, stripping down the model of ZFC, throwing out stuff, trying to get down to less set-theoretically exotic collections, trying to strip the model down to barer stuff.
This is just plain different from the talk of grounding carried out in metaphysics. In metaphysics, grounding is supposed to be a binary relation between facts, not a binary relation between models for set theory. I think it's not controversial to say that "P grounds Q" means something like "The truth of P determines or explains the truth of Q as a consequence of how the things referred to by Q are constituted from the things referred to by P," although giving a more precise account would probably involve contradicting somebody's view on the matter.
If you have models M, M' for ZFC, and M is a ground for M' in the set-theoretic sense, it doesn't exactly make sense to say that M grounds M' in the philosophers' sense, since M and M' aren't facts. You might try to say that facts about M ground (in the philosophers' sense) facts about M'. At a glance, I think this sounds quite reasonable. A bit of thought is probably required to make sure that there are no hiccups, i.e., that the resulting way of talking about set-theoretic facts (specific to a given model) doesn't violate some common expectation the philosophers have about metaphysical grounding. I have not put in that bit of thought, so I don't know if there would be any such hiccups.
Certainly, though, set-theoretic grounding does not provide a formalization of metaphysical grounding in general. I do not think it was ever intended for that purpose. An ordinary claim about metaphysical grounding involving physical objects, like SEP's example that
"The truckers are picketing" grounds "The truckers are on strike" (from https://plato.stanford.edu/entries/grounding/),
does not seem to be expressible using the set-theoretic notion of grounding at all. As I said above, set-theoretic grounding is a relation between models of ZFC, and recall that ZFC is a pure set theory, in which everything in a set. ZFC has no means to talk about individual physical objects at all, and in fact you have to modify ZFC by weakening the extensionality axiom to speak of individual physical objects. (The trouble is that, if I have a physical object A, then A has no elements since it isn't a set, and the empty set also has no elements. So A and the empty set have the same elements. Extensionality then says that A must be equal to the empty set! To speak of physical objects, better to use ZFU, which is the usual version of ZF with extensionality weakened so that you can reason about physical objects.) So you would need to modify the framework for set-theoretic grounding to even get started applying the ideas to actual physical objects, like truckers, picket lines, and strikes.
