DOF is a confusing issue because so many parameters are involved. Usually all the variables are not stated, which allows for saying confusing and wrong things about DOF. Let me give you some examples.
Wide angle lenses have more DOF. Is this correct? Yes, our experience with wide angle lenses is that we have more DOF than with longer lenses. But note the condition: the statement is true when shooting at similar distances.
Wide angle lenses have the SAME DOF as longer lenses. Yes, also true. Wide and tele lenses have the same DOF with the same framing. Can we get more DOF in an insect closeup by switching to a wider lens? No, certainly not, or we would all be using that trick in our macro work.
Crop frame cameras get more DOF. Our everyday experience is that this is true, when shooting at the same aperture. Similarly, cropping our images gets more DOF. The tricky part here is that we also get more diffraction with the crop frame camera. And it is diffraction that ultimately limits the achievable DOF. If we narrow the crop frame aperture about a stop, then DOF and diffraction are the same for the two formats. So the better answer here is that the formats give equal DOF when the crop camera is set for the equivalent aperture, which is aperture number * crop factor.
Quiz time. Which has the greater DOF?
FF camera, 100mm lens at f/8, focused to 3 meters, vs
Crop camera, 100mm lens at f/8, focused to 3 meters.
No, it's not the crop camera.
Remember, only aperture affects the DOF for the same framing (assuming the same sensor and cropping).