This is a [potentially] soft question that I feel like encompasses some pedagogical elements but possibly also advanced foundations, so I hope this is the proper forum to ask.
I'll give some background explanation to motivate where my question is coming from, so it might be easier to understand what I'm asking, in case it is not clear. Otherwise, skip to the bottom for the question itself.
Analytic geometry is a study/technique to turn algebraic problems into geometry problems or vice-versa, because problems in one space could be far easier than in the other. Here are some examples, and how I interpret these examples:
Geometry is simpler: A linear system of two independent equations must have exactly one solution. This is annoying to prove algebraically, but is completely trivial if you translate it into a geometry problem by graphing the equations: Two lines pointing in different directions can only intersect at exactly one point.
Algebra is simpler: Draw an arbitrary triangle, then draw the three medians. These three medians must intersect at a single point, known as the centroid. Proving that the medians intersect, in other words that the centroid exists, is a pretty difficult problem using only methods in geometry. Even if there exists a proof that is simple, discovering the proof by oneself seems really hard. On the other hand, although the algebra is tedious, converting this problem to an algebraic one is not only simple but makes the problem trivially easy to solve. All one has to do is write out the linear equations of the medians, and show that they have a common solution.
Complex numbers: Complex numbers can be represented in two ways: rectangular form and polar form. In rectangular form, addition is trivial, multiplication is a bit more tedious. In polar form, multiplication is trivial, addition is so difficult that I basically think about it geometrically. To me, the classic pedagogy of complex numbers, where you first learn about them in rectangular form, motivates many of the algebraic manipulations and techniques. Then once I learned about polar form, I mostly viewed the complex numbers graphically while working with them. The trigonometric angle sum formulas can be derived trivially by understanding how polar form multiplication works (geometric interpretation) and then writing out the equation in rectangular form to express it (algebraic interpretation). Neglecting either interpretation makes understanding the angle sum formula difficult.
In one sense, this seems to highlight the point that these are very different disciplines that simply converge on the same ideas using different toolsets. However, since I presume they fall under the same abstract foundations (for instance, maybe ZF axioms), they really should be one and the same.
For instance, the Parallel Postulate in geometry: "In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point." seems to be logically equivalent to the following statement in algebra: "For any $(x,y)$ not satisfying $ax+by = c$, there exists a $k \ne c$ such that $ax+by = k$." It is not immediately intuitively clear whether this axiom in geometry can be proved by the others, but the algebraic interpretation seems to be obvious to me in how it can be broken down by other algebraic axioms.
Of course, being an amateur mathematician, I'm aware that everything I've typed thus far is sort of an abstract feeling I've developed from my experiences, so it's difficult to even tell how valid my question is. Perhaps the difference isn't in the math itself, but simply the way we present them, that allows our brain to organize the ideas better and make intuitive assumptions that are otherwise insanely difficult to justify rigorously (such as the definition of a real number, or visualizing function behavior by simply graphing it)? To what extent is it just historical convention rather than objectively what's happening in the math?
If I had more time I would trace each algebra/geometry axiom down to their ZF (or whatever other system) axioms and compare them myself, but I feel like I currently lack the time to learn the nuances of these systems, compile the differences, then analyze how they are applied at a higher level. I'm hoping someone who has more expertise can help.
And so my question:
In what way do the axioms of algebra and the axioms of geometry differ such that they give us completely different intuitions and toolsets to solving problems, so much so that one interpretation can be impossibly hard while the other is trivially easy? What's going on under the hood, at the foundations level, that causes such a difference? Or is it just human intuition and not the math itself that is different?