Sony A7R III has a 42.0MP Full frame (35.9 x 24 mm ) sized BSI-CMOS sensor and features Bionz X processor. On the other hand, Nikon Z 7 has a 46.0MP Full frame (35.9 x 23.9 mm ) sized CMOS sensor and features Expeed 6 processor.
Another difference between these two cameras is that Sony A7R III's sensor lacks anti-alias (Low-Pass) filter. Removing anti-alias filter increases the sharpness and level of detail but at the same time, it increases the chance of moire occurring in certain scenes.
Below you can see the A7R III and Z 7 sensor size comparison.
Sony A7R III and Nikon Z 7 have almost the same sensor size so none of them has any significant advantage over other in providing control over depth of field when used with same focal length and aperture.