It is well known that Euler's totient satisfies $ \phi(mn) = \phi(m) \phi(n) \frac{d}{\phi(d)}, $ where $d = \gcd(m,n)$. By setting $ f(x)=\frac{\phi(x)}{x} $ this can be written as $ f(mn)f(d) = f(m)f(n) $
Have the functions that satisfy this equation been studied? They are generalized multiplicative functions.
Another generalization might be $ f(l)f(d) = f(m)f(n) $ where $l=\text{lcm}(m,n)$. The identity function satisfies this equation.