Mathematicians are engaged in a dramatically different kind of project than almost any other human discipline. Mathematical details emerge from definitions, but they appear exactly the same way for everyone else using the same definitions. And what's really surprising is how robustly those purely rational results compare to messy empirical reality. There is no good reason to believe that this should be the case!
In most other human endeavors, when something doesn't work we just work until find something else that does work, and marvel at our ingenuity. Mathematics doesn't quite give you that option. There are right and wrong answers to questions that we create ourselves, but those answers are fixed once we ask the questions, even if we didn't know what they were.
I don't think the fifth axiom is considered particularly mysterious anymore. The traditional fifth axiom clearly isn't a logical result of the first four, since it can be replaced with other parallel postulates to yield non-Euclidean geometries which are themselves perfectly workable and consistent. In fact, that section of the Wikipedia article notes that Beltrami proved the independence of the parallel postulate.
It's mysterious because it is only clear in retrospect. The fact that you can negate the parallel postulate and get a system that is still self-consistent is incredibly mysterious. This is what sets mathematics apart. We "make it up" like other human accomplishments, and yet we can't actually just make it up. If we arbitrarily defined 0^0 = π, we'd just be speaking nonsense.
It's not mysterious that negating the parallel postulate is consistent; you just have to divorce the axioms from their originally intended meanings, and realize that they apply to other objects besides those meanings.
It's kind of like this. A biologist independently discovers directed graphs, but refers to the nodes as "organisms" and edges as "parenthoods", and then spends five hundred years trying to prove that every organism has only finitely many parenthoods, thinking it must be true since it's obvious biologically! Without realizing that graphs are far more general and apply elsewhere.
Arbitrarily defining 0^0=π would not be speaking nonsense, it would just be speaking a little arbitrarily. Nonsense would be defining 0^0=rainbow
In most other human endeavors, when something doesn't work we just work until find something else that does work, and marvel at our ingenuity. Mathematics doesn't quite give you that option. There are right and wrong answers to questions that we create ourselves, but those answers are fixed once we ask the questions, even if we didn't know what they were.
One enduring mystery in this vein was whether or not Euclid's fifth axiom was actually a logical result of the first four: https://en.wikipedia.org/wiki/Parallel_postulate#History