My question by itself is hard to ask and hard to answer. I don't mean to be ambiguous, but it is one of those philosophical questions that requires a bit of specification. Let me provide some context.
Axioms are often used in two ways. The older and original concept seems to me like a top-down description. You can work with logical systems like numbers or lines on planes all you want, but when you try to backtrack your deductions, you come across axioms that can not be traced further back, much like how atoms can be not split further. Axiosm are also often used to describe assertions that systems are built from, like building blocks. I call these bottom-up axioms. That there may be no distinction between the two types is a philosophical question for another thread.
Let me use Euclidean axioms as an example. Mathematicians thought they were top-down for hundreds of years until they realized that the first four can be used independently of the fifth. Then they realized that you can "play" with axioms and basically develop novel systems that may or may not correspond to real world things. A similar process with number systems lead to the develop of modern algebra. After trying to find the most fundamental statements underlying number systems, mathematicians realized you can basically mix and match properties of operations to create new systems. But how did these mathematicians become convinced that their axioms were "true" in the first place?
For further clarification, I have no doubt that the axioms, in whatever direction, used by mathematicians do indeed describe mathematical objects very well. But I am referring to top-down axioms in my question. With this additional information, maybe my initial question is easier to understand. How did/do mathematicians deduce that their axioms were/are sufficient in describing whatever abstract object that they are pointing to?
I believe that this answer boils down to experimentation. Assert the axioms and if anything "breaks" then add more axioms, roughly speaking. Further independent study of the axioms themselves can provide more intuitive confirmation. At some point, you become so sure of the axioms that it is like you might as well define objects by their axioms. Hence, this is why top-down axioms and bottom-up axioms are used interchangeably.
If you could direct to me to any relevant texts, that would be very welcome.