What kind of "symmetry" is the symmetric group about?

Mathematically, a symmetry is a bijection of a set onto itself. In some cases, we restrict to bijections that preserve some structure (e.g. isometries, which preserve distances), but in the case of the symmetric group $S_n$ we just have a set with $n$ elements, and no additional structure to preserve, so all bijections (permutations) are allowed.

I admit to be ignorant of the actual history of the term "symmetric", but I do have a geometric explanation of why $S_n$ represents unrestricted (finite) symmetries.

First, you should ask yourself what is the special property of the (equilateral) triangle that makes its distance-preserving symmetries be $S_3$? The answer is that in the equilateral triangle there are no special relationships between the vertices, as the vertices are linearly independent in the sense that they determine a highest-dimensional object three points could possibly define, and also as they are equally spaced out (the distance between any two vertices is the same) and hence you cannot distinguish them based on distance. (It takes a little bit of slightly more rigorous and technical work to show that this fact gives you that the distance-preserving symmetries form $S_3$, which work will depend on whether you use the Coxeter presentation or the permutation definition of $S_n$; consider that homework)

For a square, one of the vertices lies in the plane determined by the other three, which means that the fourth vertex is a linear combination of the other three, so in particular the four vertices cannot possibly be equally spaced out by the triangle inequality, hence the vertices have special relationships that can be determined by distance, which is why you get the smaller $D_4$ rather than the larger $S_4$.

To get the full $S_n$ in general, you need the $n$ vertices to determine a figure of the largest possible dimension (otherwise the triangle inequality prevents the possibility of the vertices being equally spaced out), and this turns out to be sufficient for you to be able to equally space out the vertices. So, $S_4$ would be the symmetry group of distance-preserving transformations of the $3$-dimensional tetrahedron, and $S_5$ would be the symmetry group of the $4$-dimensional simplex, and so on.

This is just speculation, but I suspect that it might have something to do with symmetric functions. A function $f$ of $n$ variables is said to be symmetric if its value is unchanged for any permutation of the variables: $$ f(x_{\pi(1)},\dots,x_{\pi(n)}) = f(x_1,\dots,x_n) $$ for all $\pi \in S_n$. For example, the polynomial $f(x_1,x_2,x_3)=x_1 x_2 + x_1 x_3 + x_2 x_3$ is symmetric: $f(x_1,x_2,x_3)=f(x_1,x_3,x_2)=f(x_2,x_3,x_1)=\dots$.

Alfred Young wrote a famous series of papers about such things under the title "On Quantitative Substitutional Analysis". In the first part (from 1900), he uses the name "symmetric group" without comment, so the terminology must be older than that.