Is there a way to project one elliptic-curve element to a subgroup with certain size?
Is there a way to do something like this for elliptic curves?


Of course; if we have an elliptic curve with order $q = a \cdot b$, then we can make an element $H$ into the subgroup of order $a$ by computing $H_a = (q/a)H$.  And, we have $a H_a = 0$ (the neutral element) precisely as requested.

Now, we typically don't use elliptic curves with an order with two large prime factors; they are usually selected with an order $q = hr$, where $h$ is a small integer (most often 1, 4 or 8) and $r$ is a large prime.  We do this because that makes the ECDLP problem as hard as possible; with your order $q = a \cdot b$, you could compute discrete logs in time $\sqrt{ \max( a, b ) }$, and so we'd have to use a larger elliptic curve that we would otherwise need.