Asking what charge, momentum or energy (or other conserved quantities like QCD color) in Physics basically boils down to something that is invariant under some symmetry.

Momentum and energy may feel more intuitive, but I'm not sure they really are, especially within QM.

I'm not sure if we have any deeper explanations than these symmetries.

Btw, questions formed like "What is X?" can have this kind of problem in any domain, especially if we expect some answer that is both intuitive and provides an essentialist explanation.

For instance "What is consciousness?". "What is intelligence?", "What is the meaning of life?"

What I've come to think, is that these questions come from the same type of mistake:

As any Physicist would know, the world as described by Physics and the world as we intuitively start to understand it as small children are quite different, especially at scales far removed from our senses (like in QM or Cosmology).

Humans simply doesn't have access to the full extent of reality, nor would our brains be able to do something useful with it if we had it, since we don't have anything near the processing power to comprehend it.

What we're always stuck in, is an inner world model that is some kind of rough representation of the outside world. Now let's assume the outside world actually EXISTS, even if we don't know all that much about it. Physics is just a hint of this mismatch. If we simply let go of the assumption that there is a close correspondence between our internal model of the world and the actual world, we no longer have an obligation to form strict correspondences between object within our internal simplified simulation and the outside world.

Now we're prepared for the next step: To understand that there probably is a REASON why we have this internal representation: It's there for evolutionary purposes. It helps us act in a world. Even for concepts that do not have a 1:1 correspondence with something in the Physical world, they may very well have correspondences to aspects of the world we're simply not able to comprehend otherwise. For instance, fully understanding what "consciousness" represents (how it emerges) may not even be possible without extreme amounts of computational power (the compute part may be irreducible).

Concepts like charge are similar, except that we DO (through some advanced math) have some kind of ability to build mental models that DO (perhaps) capture what gives rise to it in the Physical world.

But it still will not map onto our intuition in a way that give us the feeling of "understanding" what it "is". It kind of feels like "consciousness is an emergent property of sufficiently large scale computational system that build world models that include themselves". Still doesn't correspond to how we "feel" that consciousness "is".

But if we simply stop insisting on full correspondence between the intuitive representation of the world and the "real" (or rather, the one represented through accumulated scientific knowledge), but instead realize that the intuition MAY still be useful, we not only avoid stress related to the disconnect, we even allow ourselves to bring back concepts (like "free will") into our intuitive world model without worrying about whether it's "real".

This provides two benefits:

1) We are "allowed" to use concepts that we know are not 100% accurate representations, and even have good reason to believe they're fairly useful simplifications of aspects of the world that ARE real, but too complex for us to grasp (like QM charge for a 5-year-old).

2) As opposed to idealists (who think the inner word is primary), we don't fall into the trap of applying those concepts out of context. Many idealist philosophies and ideologies can fail catastrophically by treating such simplified ideas as fundamental axioms from which they can deduce all sorts of absurdities.