Connecting Coordinate Geometry and Plane Geometry

What is it that allows us to take theorems proven in Euclidean geometry (i.e. with Euclid's five postulates or Hilbert's Axioms) and then apply them outside of Euclidean geometry. For example in trigonometry, where you are dealing with a unit circle in the coordinate plane, theorems regarding vertical angles, transversals, and similar triangles are openly used. I've seen proofs of these theorems in plane geometry, but I've never seen these theorems proven in coordinate geometry, and never heard of an axiom connecting plane geometry and coordinate geometry, so what is it that allows us to use our earlier theorems?

Is the Euclidean plane really the same thing as $\mathbb{R}^2$, with it's metric $\sqrt{(x_1-x_2^2)^2+(y_1-y_2)^2}$ and all? If this is the case, then it seems that we need to are taking the axioms of geometry and applying them to the space $\mathbb{R}^2$, and therefore have more postulates about the real numbers than the ordered field axioms and the least upper bound property.

Furthermore, is there an axiom connecting theorems proven in $\mathbb{R^n}$ to algebraic concepts, like the equation of a line? If we were to prove geometrically that a function's graph is a line, then don't we need some kind of axiom connecting the real numbers and geometry to state that it therefore must be true that $f(x) = mx + b$ for some constants $m$ and $b$? And vice-versa, is there an axiom that justifies using algebra to prove geometric theorems?

You are that right, the issue is usually glossed over. But it is easy to prove that the coordinate geometry of $\mathbb{R}^2$ satisfies Euclid's (and Hilbert's) axioms. Hilbert also showed that any plane satisfying his (second order) axioms is isomorphic to the ordinary coordinate plane. This is substantially harder. Such a result cannot be proved for Euclid's plane geometry, because of missing axioms.