Sony a1 has a 50.0MP Full frame (35.9 x 24 mm ) sized BSI-CMOS sensor and features Dual Bionz XR processor. On the other hand, Sony A7R III has a 42.0MP Full frame (35.9 x 24 mm ) sized BSI-CMOS sensor and features Bionz X processor.
Sony a1's sensor provides 8MP more than Sony A7R III's sensor, which gives a significant advantage in real life. You can print your images larger or crop more freely.
On the other hand, please keep in mind that Max sensor resolution is not the only determinant of resolving power. Factors such as the optical elements, low pass filter, pixel size and sensor technology also affects the final resolution of the captured image.
One other difference between these two cameras that is worth mentioning is that Sony A7R III's sensor doesn't have an 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 a1 and A7R III sensor size comparison.
Sony a1 and Sony A7R III have sensor sizes so they will provide same level of control over the depth of field when used with same focal length and aperture. On the other hand, since Sony A7R III has 18% larger pixel area (20.43µm2 vs 17.31µm2) compared to Sony a1, it has larger pixel area to collect light hence potential to have less noise in low light / High ISO images.