In other words, is there a name for this property of a function $f$: $$f(\alpha x_1,x_2,\ldots,x_n) = f(x_1,\alpha x_2,\ldots,x_n) = \ldots = f(x_1,x_2,\ldots,\alpha x_n)$$ Edit: I appreciate the answers received; however, they focus on identifying the functions that have this property rather than giving the right term (if it exists). This question is about terminology.
Even though the set of functions with this property is rather narrow, I was wondering whether there's a term or a combination of terms that would describe it. There are terms for similarly stated properties (e.g. symmetry and homogeneity/scale invariance); so I was hoping this property could be described as succinctly.
The aim of this answer is to give a characterization of functions $f$ satisfying the hypothesis that multiplying any one of the coordinated by a fixed constant $c$ gives the same value as multiplying another of the coordinates by $c.$ Our claim is that any such function is equal to some one variable function of the product of the variables of $f.$ [It is clear on the other hand that functions of this type satisfy the hypothesis.]
Given the variables $x_1,x_2,\ldots x_n.$ for each $k$ from $1$ to $n$ let $p_k$ denote the partial product $x_1x_2\cdots x_k.$ Now first begin with the value $f(p_{n-1},1,\cdots 1)$ and apply the relation using $c=x_n$ at the first and $n$th variables to get $$f(p_n,1,\ldots 1)=f(p_{n-1},1, \ldots,1,x_n.\tag{1}$$ Next begin with the value of $f(p_{n-2},1,\ldots 1, x_n)$ and apply the relation using $c=x_{n-1}$ at the first and $(n-1)$st variables to arrive at $$f(p_{n-1},1,\ldots, 1, x_n) = f(p_{n-2},1,\ldots,1,x_{n-1},x_n)\tag{2}$$
Note now that combining $(1),(2)$ we have shown that $g(p_n)=f(p_n,1,\ldots 1)$ is equal to the result of shifting $x_n$ into the final variable slot, and $x_{n-1}$ into the slot to the left of the final one. It seems clear (or could induct) that after doing $n-1$ steps of this kind, we arrive at the result that the function f we began with is actually simply a function of the product $p_n$ of the varibles. That is, $$g(x_1x_2\cdots x_n)=f(x_1,x_2,\cdots,x_n.$$