Why, in low energy situations like atomic physics, are massive particles found to be in integer number states? In quantum field theory electrons are conceptualized as quantized excitations of the quantum electron field. Generically the electron field can be in a superposition of number states. This is related to the fact that under QFT Hamiltonian/Lagrangian energy can be exchanged between electrons and other quantum fields reducing the number of electrons while increasing the number of quantum in other fields.
However, in low energy situations like atomic physics, it is overwhelmingly likely to find the electron field to have a fixed integer number of electron quanta. For example the hydrogen atom always has 1 (and not 1.2 or 1.5) electron quanta and likewise the carbon atom has 12 (and not 12.3 or 11.7) electron quanta.
Note that this in contrast to the photon field which is regularly found in photon coherent states which are superpositions of photon number states.
Why are fixed integer  states with little or no number uncertainty likely for massive quantum fields like the electron field in low energy situations?
It may be related to the fact that nuclei are found to have fixed proton numbers so charge neutrality for bound states leads to fixed electron numbers.
But this kicks the can down the road. Why don’t we see, for example, a superposition state of multiple proton and electron numbers. Or maybe out differently, why don’t we see stable bound atomic states that are superpositions of H and He atoms?
Does this have to do with the fact the electrons and protons are massive?
 A: This is a good question. It turns out to be several different questions.

For example the hydrogen atom always has 1 (and not 1.2 or 1.5) electron quanta

This is not actually true. The hydrogen atom is by definition a charge eigenstate, but an energy eigenstate of hydrogen is only approximately an electron number eigenstate. The approximation is a good one because the atomic energy scale is small compared to the mass-energy of an electron-positron pair. For heavy enough atoms (hand-wavingly, those with $Z\gtrsim 1/\alpha$), you can "spark the vacuum," and the uncertainty in particle number is of order unity. My knowledge of QCD is not great, but I believe that, for example, a hydrogen atom does have an uncertainty in quark number that is of order unity.

Note that this in contrast to the photon field which is regularly found in photon coherent states which are superpositions of photon number states.

There is a number-phase uncertainty relation. Because of this, you can't get a classical field (one whose phase can be measured) made out of fermions. To get coherence, you need to have the density of particles (per unit volume $\lambda^3$) to be $\gg 1$, but this is impossible for fermions.
For massive neutral bosons, it's possible in principle to have a coherent wave, but because of the number-phase uncertainty relation, it will have to have a large uncertainty in particle number. This means that it has a large uncertainty in energy.

But this kicks the can down the road. Why don’t we see, for example, a superposition state of multiple proton and electron numbers.

You can have this, for example, in beta decay. If you prepare the parent nucleus in its ground state, and then wait, the Schrödinger equation says that the wave function becomes a mixture of the decayed and undecayed states. (The decayed state includes the emitted beta particle and neutrino.) These states aren't uncommon in any definable objective sense, but normally when we do nuclear physics experiments, we measure the emitted beta particle as our way of detecting that something has happened. Then we get decoherence between the state of the detector that has seen a beta particle and the one that hasn't.

Or maybe out differently, why don’t we see stable bound atomic states that are superpositions of H and He atoms?

These systems have different baryon numbers. Baryon number is not only conserved, but we also don't have any observables $A$ such that $\langle \text{H} | A | \text{He} \rangle\ne0$. Therefore they inhabit different superselection sectors, and we can't make coherent superpositions of them. So there is nothing in quantum mechanics that prohibits the superposition, but the superposition doesn't have any observable consequences, e.g., you can't observe interference effects.
