In order to decide that two sets have the same number of elements (for this relationship various names have been used, e.g. equipotence, equipollence, equinumerosity), you do not need numbers or being able to count.

You just need to be able to show an one-to-one correspondence between the elements of the two sets. If an one-to-one correspondence cannot exist, then the sets have different numbers of elements.

This relationship divides then the sets in equivalence classes. If you choose a representative of each equivalence class that you use to compare to other sets to see if they have the same number of elements and you give a name to each of those representatives, you have defined the so-called natural numbers.

This is actually how the numbers originated, for humans and for many other animals.

Nobody conceived a system of axioms and then thought about what could satisfy them. That came much later and is useful only for establishing which are the essential properties of some mathematical objects. Most of the definitions of various mathematical objects as equivalence classes correspond to their real historical origin, because recognizing that some things are equivalent according to some criterion is how abstract concepts are created based on concrete things.

When you see a red apple and a red rose, you understand that they have a common property, being red, and then you name this property "red" and you can recognize the same property in other objects.

When you see 5 sheep and 5 crows, you understand that these groups have a common property, having 5 members, and the same property characterizes the set of fingers of your hand. You name this property "five" and when you see another group of things you can compare it with the set of fingers of your hand to see if it also has 5 members.

you can show two sets have the same number of elements without having an intrinsic notion of "number" - find a bijection between them, mapping every member of set A to set B and vice versa, and you know you have two identically-sized sets without doing any counting.

And for completeness’s sake, this is how the concept of "having the same size" can be, and indeed is, extended to infinite sets.