I have always felt moderately "skeptical" about the notion that Euclidean geometry as presented in a mathematics course (whether in a synthetic or an analytic form) actually corresponds to our intuitive, visual notion of flat (2d or 3d) space. By "skeptical" I of course do not mean that I actually doubt that it is true, I merely mean that I do not feel I have fully understood why it must be true, I have not fully convinced myself. Why, for instance, should a set like $\{(x,y)\in \mathbb{R}^2:y=mx+b\}$ for some $m,b \in \mathbb{R}$ correspond to the visual notion of a straight line? Why should something like $\arccos(\frac{v\cdot u}{|v||u|})$ (or however we end up defining angle formally) correspond to the visual notion of the angle between two vectors? Etc. I have never felt that I had totally satisfactory answers to these questions.
Some time ago I came across this answer on MO, explaining why topological spaces are defined in terms of open sets. The author demonstrates in a semi-rigorous way that the properties of open sets follow directly from certain intuitions we might have about "generalized rulers". Thus, from these intuitions about "generalized rulers", we can recover the formal definition of a topological space via open sets. I have been idly contemplating how one might do something similar for Euclidean geometry—demonstrate in a semi-formal way that the basic geometrical structure of $\mathbb{R}^n$ really does capture enough of our core intuitions about flat space that the picture we all have of $\mathbb{R}^n$ in our heads is really justified. But I am not sure how to do it. Certainly, since $\mathbb{R}^n$ is a model of any axiomatization of Euclidean geometry, showing that analytic geometry captures our intuitions about flat space is enough to show that synthetic geometry captures at least a subset of our intuitions about flat space.
I suspect that the "proof" here will be somewhat more complicated than the one in the linked question, as there are more intuitions to check. So I am not necessarily expecting a complete answer. But if anyone could suggest a direction in which to begin looking/thinking in order to convince myself of this fact, I would greatly appreciate it.