You're right that you can obtain the same picture of the 1.5CF cam by using the FF cam pic and crop it to the same size and every thing in the pic will be the same (except of course the subjects will appear relatively bigger in the frame for the 1.5CF). The image formed on the 1.5CF sensor is exactly the same as the one on full frame except for the crop. So as far as the original image is concerned, it's exactly the same.
The difference comes in viewing the picture and this is where DOF is different because DOF is a perception issue about what appears to be in focus and what is not.
So DOF is affected not only by the actual image image formed on the sensor but also by the magnification needed on the original image to make it into a certain picture viewing size and also by the picture viewing distance. Magnification of the original image to for e.g. 4R picture size :
Full frame : 4.2x
1.5CF sensor : 6.6x
That is why the sensor size comes into the calculation and is incorporated into DOF calculators. Try using the same focal length , same subject distance and same aperture in the calculator below to see the difference in DOF with different sensor sizes. Smaller sensors, all things equal, have shallower DOF. (One may wonder about the anomaly between the expected much shallower DOF for the very small PnS sensors and the great DOF experienced in real life. This can be explained by the fact that small PnS sensors' shallow DOF is usually overwhelmed by the short focal length used) :
http://photoinf.com/Tools/Don_Fleming/Depth_Of_Filed_Calculator.html
This is usually explained with the term "Circle of Confusion".
Take a look at the 2 pictures below and one would perceive that the 2nd picture (cropped picture to simulate taken at the same focal length, same light condition, same aperture, same distance etc. except on a smaller sensor size) has a shallower DOF even though it's the same picture. Notice the small leaves in the background (at the top left hand corner of both pictures or other places). In the original picture, the leaves which appears much smaller also looks much sharper than in the cropped picture. That's the effect of different magnification needed to achieve the same picture viewing size.
Original picture (downsized to 800x533):
Cropped and then downsized to 800x533 to have the same viewing size as the above picture :