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.