What is the name for functions like this: $(x,y,z)\mapsto (f(x),g(y),h(z))$? For $n=3$: $$(x,y,z)\mapsto(f(x),g(y),h(z)).$$
For arbitrary $n$: $$(x_1,...,x_n)\mapsto\,(f_1(x_1),\dots,\,f_n(x_n)).$$
I don't need the answer for mathematics' sake, but to suitably name a multivariate function in some computer program that conforms to this model.

I would refer to this as a type of (direct) product function. Namely, you have functions $f_i\colon\Bbb R\to\Bbb R$, $i=1,2,\dots,n$, and you have the function $f_1\times\dots\times f_n$ given by
$$(f_1\times\dots\times f_n)(x_1,\dots,x_n) = \big(f_1(x_1),f_2(x_2),\dots,f_n(x_n)\big).$$
