The axioms of group theory are not assumed to be true. They should be taken as the definition of the term, "group". That makes them true, literally by definition, so long as you are doing group theory, but they may fail to be true when you find yourself working with semigroups (or metric spaces, or some other structure). They are not based on intuition, but on experience, as objects that satisfy them keep on coming up wherever Mathematics is done or applied.
The axioms of Euclidean geometry were assumed to be true. It was assumed, based on intuition, that they applied to the physical world, and that things could be no other way. Nowadays, we know things can be another way, both in abstract Mathematics and in (mathematical models of) physical space, and the situation is more like that in group theory: the axioms of Euclidean geometry are taken to define Euclidean geometry. Alternatively, the axioms of Euclidean geometry can be proved to hold by bringing in Cartesian co-ordinates and arguing via analytic geometry (but then you're bringing in a whole other troop of axioms...).
The axioms of set theory, I think, come closest to the model you propose. They can be taken to define the subject matter of set theory, but the reason the current axioms are chosen, and not some others, is (I think) that the current axioms seem to capture the intuitions that mathematicians have built up about sets. Their history has been a mix of logical deductions and "obvious truths". Occasionally, logical deduction has shown that some "obvious truth" actually leads to a contradiction (I'm thinking in particular of Russell's demolition of Frege's conception of sets), and that has led us to fall back on somewhat less obvious truths. Like I said in the comments, it's complicated.