Double discrete logarithm on elliptic curve
However, in the case of elliptic curves, $h^x$ is interpreted as the point 
  $h$ added to itself $x$ times, resulting in curve point, $[x]h$


That is incorrect; as you had stated previously:


  $h \in \mathbb{Z}_p^*$ is an element of order $q$.


That is, $h$ is not a point on the elliptic curve; instead, it is a member of a finite field, and so $h^x$ is interpreted as $h$ multiplied to itself $x$ times (using the field multiplication operation).


  The folklore is that any cryptographic scheme based on a modular multiplicative group can be translated to an additive curve group.


One has to be careful; there are schemes such as SRTP that really rely on a modular field (rather than a modular group).

However, even in those cases where things can translate nicely (which are actually most of them; exceptions are comparatively rare), you do have to remember that, while the actual group operation (modular multiplication) translates into point addition, the operations on the exponents (which are done modulo $p-1$ in the modular case) translate to integer operations modulo $q$ in the Elliptic Curve case.