From an optical point of view, light does not bend differently just because you put a differently-sized rectangle somewhere in its path. Or to put it another way, if you cut the edges off your sensor, that won’t alter the image on the remaining area of the sensor.
Only in the sense that you generally use a smaller sensor because you want your camera to be small.
If you take a full frame SLR and attach a 100mm f1.8 lens to it, you’ll get a shallow depth of field. Now crop that image down to an area of the sensor corresponding to the size of a phone sensor, and the cropped image will have the exact same depth of field.