Why a transistor (MOS) changes the output current (Ids) with changes of a load resistor?
Your question suggests that you think it is a property of the transistor that Ids changes with the value of the load resistor.

However you disregard the fact that it also depends on HOW you use the transistor. In your circuit, assuming the transistors are NMOS (when using PMOS, no current would flow), the transistors appear to be connected as "diodes" and in this configuration the MOSFET sort of behaves as a normal silicon diode although with a less steep current increase over voltage. So the high dependence of Ids is to be expected for the circuit you show.