When doing macro photography, is there greater depth of field with a macro lens or a telephoto lens using an adapter?
The same principal parameters lead to the same kind of image. However, a tele lens with extension tube might be working outside of its design parameters at least at wider apertures. With an extension tube, the real image is picked off way behind the lens' expectation and is constituted of light from locations the lens does not expect to be able to focus on. It is prepared to turn them into nice bokeh, not sharp images. So any corrections the lens may make to field curvature and aberrations are likely to focus on the wrong imaging plane. Also whatever measures the lens takes to keep an even light spread at all apertures may be thwarted. And if some light reaches the tube at wider apertures instead of making it to the sensor, it will be likely to reflect to some degree at such an oblique angle and cause loss of contrast.
So you should not be surprised if the total field of view does not have uniform focusing distance, sharpness, light yield and it may well be that depth of field is not entirely consistent across the image. Your chances tend to improve as the aperture narrows down, and with respect of depth of field in macro photography, a narrow aperture usually is called for anyway.
So in practical terms, the difference should be small as long as the aperture is not wider open than the optical situation would accommodate (you can try finding out with a constant white/grey subject which aperture openings are still fully effective for getting more light to your image).
And any lens constructed in a manner to deliver "interesting" bokeh of any kind may be problematic for picking a focusing plane it doesn't expect.
Other than that, it's mostly a case of using a lens in a manner it has not been designed for in practice but which it should serve well in theory. But if theory were good for everything, we'd not need lenses with a dozen elements either.
So basically, you can expect about the same parameters in operation, but less consistency and quality than with a lens designed for the actual operation, distances, and imaging plane of a macro lens.
Magnification and Aperture control depth of field.
For the same magnification and aperture, the depth of field will be the same whether or not you use a dedicated Macro lens or a telephoto with extension tubes.