Find all $f(x)$ such that $f(gcd(x,y))=gcd(f(x),f(y))$

A combination of two solutions is again a solution. This does not help much with your class of functions, but wait.

Consider any multiplicative function that maps primes to a permutation thereof. Say, $2\mapsto3$, $3\mapsto2$, and the rest of primes map to themselves. This would do as well.