According to Wikipedia, the ergodic hypothesis is the assumption that

all accessible microstates are equiprobable over a long period of time.

My question is about the precise meaning of "accessible" here.

Consider the microcanonical ensemble for a thermally insulated system (say an ideal gas of $N$ particles and $E$ units of energy, confined to a box of volume $V$). As far as I can tell, the only thing that "accessible" means here is that the energy is constant, so when computing spatial averages we only average over the states of a given energy.

But why do we only worry about the energy here? Why not worry about only averaging over the states of constant momentum, or angular momentum, or any other quantity that could potentially be conserved?

(I realize, by the way, that momentum is not conserved in an ideal gas confined to a box, but surely there are thermodynamic systems where there is some conserved quantity other than energy, right?)

What exactly is so special about energy here? And are there situations where "accessible states" refers to more than just all the states of a certain energy?