Yes—if you compare images shown or printed at the same size, a crop-sensor camera will appear to give more depth of field at the same f-number. But this is not because the lens becomes optically different just because it’s labeled “full-frame.” It’s effectively the same result as using that same lens on full frame and then cropping the image down to the smaller sensor’s field of view.

So in practical terms: same lens, same aperture, smaller sensor framing means more apparent depth of field in the final image compared with full frame using the full image area. The tradeoff is that the smaller sensor captures a narrower field of view.