Maybe this isn't quite general enough to help, but perhaps some material from Mathematical Cognition might be helpful in drawing out a distinction I think you want to be making?
In both live Mathematics research and Mathematics education, we often talk of there being two kinds of numbers.
The first kind of number that we talk about is when we have some collection of things as a quantity. For example, brains fairly quickly in life (and early in development) know to distinguish between one of something, two of something, and many of something. If I have a bunch of sweets in one pile on the table and another bunch in another pile, we can often say that we can tell just by observation and comparison whether there are more sweets in one pile over the other.
This is a broad stroke interpretation of what we mean by the "Cardinality" of a set of things, and our standard measure of cardinality, of "how many", is to use the Cardinal Numbers. Cardinality is abstractly understood - whenever we say that we can take any two sets and can functionally put them in "one-to-one correspondence" with each other, then we say that they have the same cardinality.
The second kind of number is more abstract than that, and takes a bit more teaching for people to grasp. This kind of number is what we are pointing to when we work through a sequence of names of numbers in succession. At school we are taught to "count" by working through those names in order - we go "One, Two, Three, Four, Five...". Each of the points in the sequence is understood to follow after the other, and as we are soon taught, you can "add one" to any element of this sequence to get the next element in that sequence.
This roughly speaking is what we are trying to use to demonstrate our understanding of some ordering of a sequence, and we call this measure of the type of ordering involve the Ordinal Numbers (with each number occupying a position in the sequence). To understand how to get at Ordinal "twenty five thousand, two hundred and thirty six", for example, we don't need to go out into the world and find some set of 25236 things in order to show that we have correctly understood this - we can demonstrate a familiarity with what it means to be the successor of 25235.
Now, there can be some nuance to how we conceptually use ideas of "numbering" and "numbers", because in English (at least), we often assume that measuring "how many" and measuring "how to order" are functionally the same. This is because when we're young, we're taught to use the Cardinality of Ordinal sequences to help us work out exactly how many there are in any given set of objects. This object is "one", this object is "two", this object is "three"... And, of course, through demonstrating little principles like how addition is similar to repeated addings of one, we show how the ordinal sequence of numbers can be used to give us more rich subdivisions of cardinal quantity than our brains naturally jump to themselves ("how many is ten thousand", for example), and also some neat cognitive tricks we can use to get those quantities using ordinal technology.
But the two ideas do importantly come apart. For example, even if we think there is some evolutionarily advantageous basic "number sense" of cardinality across human cognition, different cultures and societies form different models of ordinal sequences, and different people seem to have a better or worse time learning to grasp and use these sequences well. One plausible suggestion is that there are similar brain functions used in both concepts of number, but that the Ordinal one shares more in common with the language processing mechanisms of our brain, while the cardinal one ties closer to our image perception and object recognition.
If that is the case, then while it seems like Cardinal numbers can roughly reduce to being patterns of human brains, Ordinal numbers might be more of a social protocol - an abstract pattern of thought that is built up across a community of mathematics practitioners rather than just for any one of us. But that is just a theory (...), and certainly wouldn't be considered philosophical canon; if anything, a philosopher of science should be leaving this question to the psychologists to answer, rather than trying to make specific headway on it themselves.