Fibonacci sequence is used, for example, in the reproduction of rabbits.
What are the axioms that the sequence [Fibonacci sequence] is based on?
The first two terms are fixed. Every term after the second is the sum of the previous two terms. Typically, the first two terms are $0,1$ or $1,1$ (with the latter option being far more common in my experience). You could have gotten that by typing "Fibonacci sequence" into google, so I am probably not understanding what you are looking for.
Maybe I ask it in this way:
are the sequences, for instance Fibonacci Sequence, in mathematics based on Peano Axioms?
Is there another axiom, except it, that sum based on... (a) in general? (b) on the particular sequence - Fibonacci Sequences?
Perhaps reviewing the definition of a sequence would help: Given a set $X$, a sequence in $X$ is a function from the set of Natural Numbers to the set $X$.
So, if you are looking for the set of axioms required so that definition makes sense, then you need a base set theory (so that discussion of a set $X$ makes sense). Typically, this is the Zermelo-Fraenkel axioms of set theory. You need axioms for the Natural Numbers. The Peano axioms are a good set to work with, but there are others that also describe the Natural Numbers.
Is this what you are looking for?
ZF axioms offer the basic axioms needed for set theory. The Peano axioms offer the basic axioms for the construction of the Natural numbers. The Fibonacci Sequence is defined as:
$f:\mathbb{N}\to \mathbb{N}$ by $f(0)=f(1)=1$ and $\forall n>1, f(n) = f(n-2)+f(n-1)$.
You may not even need the ZF axioms. The Peano axioms may be enough.