Update/Partial Answer (see edit history for original post)
After reading comments, checking the linked post, and working on the problem a bit more, I realized that there is a straightforward definition for tangents of open sets (of Euclidean $n$-space) which does not require any kind of differential structure. This occurred to me when I noticed that the sets David K mentioned for which the line intersecting a set at exactly one point is a tangent line happen to be convex sets.
Here's what I've come up with:
Definition (convex set): Let $U\subseteq\Bbb R^n$. $U$ is convex if and only if for every line $L\subset\Bbb R^n$, the intersection $U\cap L$ is connected (w.r.t. the Euclidean subspace topology on $U$.)
Note: "line" is treated as primitive. As such, every model of geometry (assumed to be an incidence structure, for the time being) has different convex sets. This is where the interplay between the geometry and topology becomes important: the same topological space can have more than one geometry defined on it. In what follows, we will need both a topology and a geometry in order to define "tangentness."
Definition (exterior tangent of a convex open set): Let $U\subseteq\Bbb R^n$ be a convex open set. Call $V\subseteq\Bbb R^n$ an exterior tangent of $U$ if and only if $\emptyset\subset\overline U\cap V\subseteq \partial U$. (Alternatively, $V$ is tangent to $\overline U$; this is a matter of convention, pick whichever one is most standard.)
Definition (interior and exterior tangent of an open set): Let $U\subseteq\Bbb R^n$ be an open set and $V\subseteq\Bbb R^n$. Call $V$ an exterior tangent of $U$ if and only if there exists a convex open set $S\subseteq U$ such that $\emptyset\subset \overline S\cap V=\overline U\cap V\subseteq \partial U$.
(continued) We call $V$ an interior tangent of $U$ if and only if it is an exterior tangent of $\overline U^\complement$.
Definition (tangent of an open set): Let $U\subseteq\Bbb R^n$ be an open set. A set $V$ is tangent to $U$ if and only if it is an exterior tangent or interior tangent of $U$.
From here, we naturally get tangent lines, planes, hyperplanes, etc. by way of "a tangent line is a tangent that is also a line," "a tangent plane is a tangent that is also a plane," etc. - with each of these ("line," "plane," etc.) being defined by the geometry on $\Bbb R^n$.
We also get "point of tangency" - and "set of tangency" when the tangent is coincident with the boundary at multiple points - in an obvious.
From here, I have two new questions:
- Does the above definition work? If so, can it be refined?
It seems obvious to me that the set of "tangents" of a given set, as defined above, is inclusive of conventional tangent spaces (along with some other stuff.) Are there sets satisfying the above criteria which are definitively not tangent? (see note for additional clarification)
- How do we extend this notion of "tangentness" to non-open sets?
The suggested definition leads to a clear distinction between two kinds of tangency: there are the tangents of open sets, which I have defined above, and the tangents of... not open sets. As an example of the latter kind of tangency, consider a tangent line to surface $S\subset\Bbb R^n$, with $n>2$. Obviously $S$ is not an open set, so the definition does not apply. To make matters worse, even if $S$ were an open set, there are many cases in which $\partial S=\emptyset$ where we still want $S$ to have tangents.
In the case that $S$ is instead the boundary of some open set $U$, we might get away with saying that "$V$ is tangent to $S$ iff $V$ is tangent to $U$" - for example "for any differentiable function $f:\Bbb R\to\Bbb R$, the line $T_pC=\{(x,y)\in\Bbb R^2:f'(p)(x-p)=y-f(p)\}$ is tangent to the curve $C=\{(x,y):y=f(x)\}$ at the point $(p,f(p))$ because $T_pC$ is tangent to the open set $\{(x,y)\in\Bbb R^2:y<f(x)\}$ at $(p,f(p))$." - but the existence of enumerable "wild and wacky manifolds from the nightmare realm" makes me reluctant to suppose that every set with tangents is the boundary of an open set.
Note: assuming the Euclidean topology and geometry, the suggested definition of tangent includes sets which are not "differentiable" in any meaningful sense and/or which intersect the boundary of the open set at an angle. For example, the curve $C=\{(x,y)\in\Bbb R^2:y=\vert x\vert\}$ would be tangent to the $(0,-1)$-centered open disk of radius $1$ at the origin. I don't see this as a problem, since we need only specify "tangent line" or "smooth tangent curve" or "tangent curve that has a derivative at the point of tangency whose derivative at that point is equal to the derivative of the boundary of the set to which it is tangent at the point of tangency" to get back to analysis-land. I'm more worried about a line which intersects a sphere being "tangent" to the sphere because I missed something.
