An axiom is something that seems to be self-evidently true. It cannot be questioned or argued against. The simplest example of this is the logical law of identity, that A is A. This also implies that Non-A is Non-A, and A is not Non-A.
It's interesting how you went from "logical law of identity" to "this also implies." I don't see how Non-A is Non-A is axiomatic. It can be questioned and argued against.
It is further interesting to mix the law of logic and this axiomatic self. With the Self comes the Non-Self, as required by the law of identity. In fact, this may be the solution to solipsism itself, because for the Self to exist, the Non-Self has to exist as well, meaning there is more that exists than simply your own mind, as posited by solipsism. This, of course, doesn’t address BiaV. There are also other matters to discuss, such as whether there is just the Self and Non-Self, or if both arise from something else, though this is beyond arguing the self as an axiom.
TLDR – “I exist” is axiomatic.
Agree that "I exist" is axiomatic.
Disagree the Non-Self has to exist as well, and for sure contest the notion that there is more than exists than simply own mind.