Does (stray) capacitance depend on voltage?
Should the stray capacitance be dependent on the voltage difference?

No, it shouldn't; it's a physical property related to physical dimensions and the electric permittivity of the materials.

The electric field being higher is immaterial.


  capacitance is entirely independent of the voltage


Correct with one caveat - certain parasitic capacitances associated with reverse biased PN junctions will vary their capacitance with applied voltage but, this is because the depletion layer grows and, in effect, the gap between the "plates" increases thus lowering capacitance.