How does the security of Elliptic curve compare to normal discrete logarithm?
As far as I know the best choice is a 'safe prime' with $P=2q+1$ with $q$ a prime as well.

This is the best choice for a given size of $P$, but not for a given size of $q$. See this.

This discrete logarithm can be solved in $\mathcal{O}(\sqrt{q})$ with q the biggest prime factor (with Pollard's (Rho) algorithm).

Essentially yes (minor caveat: $\mathcal{O}(\sqrt{q})$ is not effort, but the number of multiplications of integers of size $P$, with $P&gt;q$, hence the effort grows faster by a factor at least $\ln P\,\ln\ln P$). That the DLP can be solved with such method and effort does not imply that such method or effort is needed. And if $P$ is a safe prime, there are methods (including the Number Field Sieve) requiring less effort. Again, see this.

Do (DLP in a subgroup of an appropriate Elliptic Curve on one hand, of $\mathbb Z_P^*$ on the other hand) have the same solving time of $\mathcal{O}(\sqrt{q})$ (group operations, where prime $q$ is the order of the subgroup)?

Yes, when using Pollard's Rho algorithm. That algorithm is believed optimum in the Elliptic Curve case, and for $P$ large enough in the $\mathbb Z_P^*$ case.
No, when $P$ is a safe prime (and large enough to make the DLP non-trivial), and using the Number Field Sieve to tackle the DLP in the subgroup of $\mathbb Z_P^*$.

Note: I don't know that the Number Field Sieve can be used to solve the DLP in an appropriate Elliptic Curve (sub)group; and it would come as a huge surprise if it was more efficient than Pollard's Rho algorithm.