Similar to the Poincaré disc for hyperbolic space, is there a bijection from $\mathbb{R}^n$ into, say, $[-1,1]^n$, while any paraxial orthotope in $\mathbb{R}^n$ remains a paraxial orthotope after the projection, i.e. lengths are distorted but not angles as long as they are orthogonal to the base of the space?
Now, I'm pretty certain the answer is "yes" and something as simple as
$$f(\mathbf{x})=\sum_{i=1}^n \frac{x_i}{1+|x_i|}$$
does exactly that. The question is, does it really? And does it have a name so that I can look up its other properties?
You may want to write $(-1,1)^n$ instead of $[-1,1]^n$. The image of $\mathbb R^n$ under a homeomorphism cannot be compact.
Yes, for any collection of $n$ homeomorphisms $f_i:\mathbb R \to (-1,1)$ the product map $F(x_1,\dots,x_n)=(f_1(x_1),\dots, f_n(x_n))$ provides a homeomorphism of $\mathbb R^n$ onto $(-1,1)^n$ with the stated property. It's not about angles as much as about preserving product structure of subsets: $\prod A_i$ goes to $\prod f_i(A_i)$.
I do not find much similarity with the Poincaré disc model for hyperbolic space, which involves a conformal map, i.e., on that preserves all angles.
The term "product map" is the best description I know of this thing; it's used, e.g., here.