DoF is a function of aperture and magnification.
That's a common misconception. For a given camera, depth-of-field basically is a function of distance, focal length, and aperture.
Distance and focal length cancel each other out (for practical intents and purposes) only when DOF is small in relation to the distance, i. e. with telephoto lenses (at long distances) and in macro photography (at large magnifications). They do not cancel each other out in general.
I am thinking about getting the Canon TS-E 45 mm and using it plus a 1.4× extender. [...] but what happens to depth of field in the process?
Depends on what you're trying to achieve. Will you use the extender to increase magnification (at the same distance) or to increase distance (for the same magnification)?
If I were to use the TS-E 45 mm + 1.4× (= 63 mm) at f/2.8, will the results match the EF 24-70L zoom set at 63 mm at f/2.8 in depth-of-field and relative intensity of blurring?
No. They will basically match the results of the 24-70 mm zoom lens, set to 63 mm and f/4. I said 'basically' because the actual results may still differ due to differences in the lens designs and residual aberrations.
Regarding aberrations ... tele converters usually work best when used to make long lelephoto lenses even longer. Typically they do not go very well with wide-angle or standard lenses. However shift-and-tilt lenses have an increased exit-pupil-to-image distance (to make room for the shifting and tilting mechanisms) so they will work somewhat better on tele converters than regular lenses of the same focal length would—at least as long as they're neither shifted nor tilted. When shifting or tilting, image quality on a tele converter may or may not break down, depending on the particular lens, converter, and degree of lens adjustment. If someone is getting good results using a TS-E 24 mm on a 1.4× converter it doesn't necessarily mean that results will also be good with the TS-E 45 mm lens on a converter. in any case, it means using the equipment outside the intended scope of application. So try it out before purchasing!