lens - Is the angle of view actually different on APS-C?

A 50mm prime lens, whether APS-C or Full-frame is designed around the 35mm format. As such, is the angle of view actually different on a DX/APS-C crop sensor or are we just seeing less of the full frame angle of view?

I think that the angle of view, depth of field, and compression are the same as on an FX/Full frame but we are seeing only middle, correct?

If so, then isn't it the case that a 50 1.4 is giving the same DOF on full-frame as DX but because we are seeing less, we have to stand further away, thus the "increase in DOF" on a DX?

More importantly, and the thrust of my question, is that the perspective/compression is the same regardless of crop factor on 35mm format lenses, right?

Answer

In short, yes, it is actually different, but only because the angle of view is by definition measured to the edge of your view — which is the edge of the sensor, which is (again by definition) different on APS-C.

Your statement

I think that the angle of view, depth of field, and compression are the same as on an FX/Full frame but we are seeing only middle, correct?

is completely correct (with that caveat). But, "is the angle of view actually different on a DX/APS-C crop sensor or are we just seeing less of the full frame angle of view" is meaningless, because it's called angle of view because the view is what's measured. One could measure the angle to the edge of the projected circle of light from the back of the lens, but that's not particularly useful since it's not recorded. The exercise in my answer to What is "angle of view" in photography? should make this all clear.

And you're right about the depth of field, too. More on that at Why does a bigger sensor lead to a shallower depth of field? and Can a smaller sensor's "crop factor" be used to calculate the exact increase in depth of field?.

And, you're also right about the effect on perspective: that's solely a matter of where you stand. That's covered at What is background compression?, and to quote ex-ms from an answer there:

If you take a shot with a wide-angle lens and crop down to the very centre, you would have the same background compression as if you took the shot with a telephoto lens.