Let $P$ denote a polyhderal set of $x$ values in $\mathbb{R}^n: \{x: Ax \leq b\}$. Let $Q$ denote the projection of $P$ onto $\mathbb{R}^{n-1}$ (i.e., $Q \subseteq \mathbb{R}^{n-1}$).
Do the sets $P$ and $Q$ contain the same values of $x$ in $\mathbb{R}^{n-1}$? That is, for $x = (x_1, \ldots, x_n)$ in $P$, does $Q$ contain the same $(x_1, \ldots, x_{n-1})$? In other words, are $Q$ and $P$ the same set except for 1 dimension (the $n$th dimension)?
If so, is there a name of this property, or is this inherent in the definition of a projection?
A very common definition of "projection operator" is "idempotent operator". This means that after projecting once, repeat projection does not move the point.
One problem with your statement "projecting onto $\Bbb{R}^{n-1}$" is that you don't restrict to particular choices of $\Bbb{R}^{n-1}$. There are infinitely many choices of $\Bbb{R}^{n-1}$ as a (potentially affine) subspace in $\Bbb{R}^{n}$. A projection to one of these subspaces needs to define which one is the image.
For instance, in the plane, one can project $(x,y)$ onto the line $x = 1$ by $(x,y) \mapsto (1,y)$. Then the set of $y$-values is exactly preserved by this projection. Notice that it isn't the first several coordinates that are preserved. We could renumber our coordinates so that the leading coordinates were fixed and only the last coordinate was projected away, but we would have to say so. (This is a variation of projection onto the second coordinate. You should understand that there are projections onto any subset of the coordinates. Almost all of these modify coordinates other than the last one. In $\Bbb{R}^n$, there are $n$ choices of one coordinate to "forget" by projection onto a subset of the coordinates. Only one of these preserves all but the last coordinate.)
Additionally, a projection need not be a coordinate projection. Consider the map $(x,y) \mapsto (x+y+1,-1)$. This map is idempotent, so is a projection. None of the coordinates is preserved in this projection.
So the property you are wanting is "projection onto the first $n-1$ coordinates". This means a map $$ \Bbb{R}^n \rightarrow \Bbb{R}^{n-1}: (x_1, \dots, x_{n-1}, x_n) \mapsto (x_1, \dots, x_{n-1}) \text{.} $$