1) All information available to me suggests that the "Schrödinger
pressure" is ordinarily called _degeneracy pressure_.
I get 40 hits on the former as opposed to 16,000 hits on the latter.
I recommend the latter as being more descriptive.
2) Degeneracy pressure (which is really not much different from any
other "kind" of pressure) shows up in many places, including atoms,
metals, neutron starts, et cetera.