In the case of the hydraulic lift (pictured) according to the pascals law we say that the pressure on $A_1$ is the same as $A_2$. In here do we assume that the $d_1$ value involved is so small that the pressure variation has no effect?
First, the "pressure variation" is actually called "hydrostatic pressure."
According to Wikipedia, Pascal's Law "states that a pressure change at any point in a confined incompressible fluid is transmitted throughout the fluid such that the same change occurs everywhere." The hydrostatic pressure is actually still there. The pressure at the bottom of the hydraulic lift is a bit greater than the pressure at the top because the bottom has to support the weight of all the fluid above it. What Pascal's Law means is that the additional force, which increases the pressure, is distributed evenly to all parts of the fluid.
Therefore, the absolute pressure $P$ is different in different parts of the fluid, but the increase in pressure $\Delta P$ is the same throughout the fluid.