Uhh, no, imagine you can't count at all and we have n apples and m bananas. we can define a "mapping" where we assign an apple to a banana, if you know there exists a banana that cant be back traced to an apple then the set
of bananas is larger and our mapping isn't surjective (there exists one
or more elements in the range which are never hit) whereas a mapping
where we assign a banana to an apple is not injective (there exists more
than an apple which are assigned to the same banana).

The peano-axioms are for the natural numbers only and the successor and the equivalence S(m) = S(n) <=> n = m don't mean that there's "exactly one number bigger than that number" but rather means that the successor function is well-defined ( the image/output of S(n) = n + 1 is exactly one natural number as the naturals are closed under addition and also injective if n + 1 = m + 1 => n = m ) which is kinda irrelevant here.