I read about the axioms of euclidean geometry.
How is analytic geometry rigorously defined? What are the axioms? And most important: How to prove that all the results proved in analytic geometry are indeed the same things as euclidean geometry? How to show that they are equivalent? For example: How to show that $y=x$ is indeed a line and not a curve?
Analytic geometry is not generally axiomatized in the way you seem to be asking.
Instead, one proceeds as follows. First, the real numbers $\mathbb{R}$ are axiomatized, as you learn in any advanced calculus or real analysis course. Second, the Euclidean plane $\mathbb{E}^2$ is defined as the Cartesian product $\mathbb{R} \times \mathbb{R}$, and lines are defined as solution sets of linear equations $ax+by=c$, and angles are defined using the law of cosines, and so one… Third, once all the basic concepts of Euclidean geometry are defined, the axioms of Euclidean geometry are proved as theorems.
You can see this procedure carried out in several modern geometry textbooks, for example Hartshorne's book "Geometry: Euclid and Beyond".