Identical equations for multiplicative functions

We examine identical equations for multiplicative functions and certain special cases, such as totients and quadratics. We confine ourselves to identical equations expressing the value $f(mn)$ (or the value $f(m)f(n)$) nontrivially in terms of the values $f(m/a)f(n/b)$ and $f(mn/(ab)),$ where $a\mid m$ and $b\mid n,$ and holding for all $m$ and $n.$ Particular attention is paid to Busche–Ramanujan type identities. We characterize all functions that satisfy the identical equations. Quasi-multiplicative functions are central to this discussion.