Morality is global
A strange aspect of the literature on metaethics is that most of it sees morality as a local phenomenon, located in specific acts or events (or people or outcomes). I guess this goes back to G.E. Moore, who asked what it means to call something 'good'.
That's not how I think of morality. The basic moral facts are global. They don't pertain to specific acts or events.
Here, morality contrasts with, say, phenomenal consciousness. Some creatures (in some states) are phenomenally conscious, others are not. Intuitively, this is a basic fact about the relevant creatures. Hence it makes sense to wonder whether one creature is conscious and another isn't, even if we know that they are alike in other respects. With moral properties, this doesn't make sense. If two events are alike descriptively, they must be alike morally.
Moral properties are more like mathematical properties. My desk has the property that the number of pens on it is prime. Any desk that's like mine in all non-mathematical respects also has this property. If the metaphysics of maths had gone the way of metaethics, there would be an extensive debate about the nature of such properties: can they be identified with "natural" properties? If so, is the identification a priori or a posteriori? If not, what explains the supervenience of mathematical properties on natural properties?
In either case, I'd say that what explains the "weak", intra-world supervenience is that the properties are derivative. My desk has the property that the number of pens on it is prime because (a) there are three pens on the desk, (b) if there are three things then the number of these things is 3, and (c) the number 3 is prime. (a) is a non-mathematical, "natural" fact. (c) is a mathematical fact. (b) is a general bridge principle linking the two kinds of fact. Likewise, if an act is morally good then this is always because it has some non-moral properties for which we have a general bridge principle linking those properties to the moral domain.
What explains the "strong", cross-world supervenience, in either case, is that we consider the relevant bridge principles (as well as purely mathematical/moral facts like (c)) to be metaphysically necessary.
Is my desk's mathematical property a natural property? Arguably yes: all it takes for a desk to have the property is to have three (or two, or five, etc.) pens on it. Indeed, if properties are individuated intensionally, then strong supervenience entails identity. But that's not the end of the story, if we're seeking a metaphysics of maths. It's not even the beginning of a story. It's beside the point.