In sciences there are basic principles, which are the distillates of observation.
This is a somewhat arbitrary application of the word “axiom” in distinction from principle. In math, logic and philosophy, an axiom is a
starting point that must be
assumed to proceed. The “basic” or identifiable quality of an axiom is that there does
not exist a proof from which, as a claim, the axiom can logically follow. It must be presumed in order to proceed.
Basic principles are the guidelines which are used to guide or superintend a process (logic, science, etc.) and are implicit in the methodology of that process.
You are quite correct that science claims to not rely on axioms. However, that is only true if science restricts itself to making conclusions by following those methodological principles. Your unstated assumption or axiomatic belief, on the other hand, which you correctly point out is not one made by science proper, is, however, implicit in the way
you criticize theological claims.
The axiom you attempt to covertly apply, despite your denial, is that all non-physical realities are subject to the methods of science for validation. That position is only tenable after invoking an axiomatic belief that reality must be purely physical in nature. This is an axiom because it cannot be demonstrated or verified by methods of science that must, in principle, only deal with physically detectible and measurable entities or “facts.”
Theology, ethics and other endeavours could very well have their own principles of verification that need not be subject to scientific methodologies for verification because theological or ethical claims are not essentially claims about nature. They may be verified by consistency, efficacy, revelation, historical support, personal experience, meaningfulness, explanatory power, or a number of other means.
Of course, if you wish to take it as axiomatic that all reality is and can only be
physical in nature, then you are constrained, by that starting point to discuss only matters of physical reality that can be verified by scientific principles. Anything you have to say about non-physical possibilities is to extend yourself beyond your own self-imposed limitations.
There is no reason to think the rest of us need to abide by your dogmatic guiding principles built upon a materialistic assumption.