The distinction you are drawing has actually occurred on two levels from the late 19th century through to the present. There is the level of discarding irrelevant details of an object to study the structure that remains (which is a fundamental abstraction that leads to the ideas of number and so on) and there is the level of studying how an object transforms rather than observing it "at rest".
In the following I use variations of the phrase "satisfy the axioms "as a shorthand for "satisfies a typically multi-layered collection of sets of axioms and also the definitions of a particular type of mathematical object".
First we don't care what objects are made out of, we only care that they satisfy the axioms of the system we wish to use to model them. An example: I don't care if the group I am studying is made of rotations of some object, I only care that the collection of its elements has the structure of a group (and satisfies some set of relations about which I know enough to arrive at my desired conclusions). In this sense, we are implementing the first layer of abstraction: disregard the details of implementation and pay attention to the structure that remains. The study of category theory pushes this idea much further: we stop caring about which set of axioms each object satisfies and search for patterns common across different collections of objects where each collection contains objects satisfying a particular collection of axioms. At that level, the objects need not have implementations as rotations or whatever, so we can't really "see" elements of the objects. Instead, we see the functions between pairs of objects and study the patterns in these functions, which conveniently segues to the second idea.
The study of the dual of an object satisfying a collection of axioms is the study of functions from the object into some other object. Although that article speaks of dual vector spaces, the idea of duals extends further. For instance the study of automorphisms and homomorphisms of algebraic objects is the study of the analogous dual in this other setting. Desarges' Theorem is about a duality in Euclidean geometry which (in a way more precise than I will write here) shows that theorems in geometry are unchanged when you swap the occurrences of "point" and "line". In the dual setting, the set of functions which fold, spindle, and otherwise deform the object is interesting, because we restrict to functions that preserve some property of the object. A measure is a function from (some) subsets of a given set to real numbers that converts nesting to ordering (small sets to small numbers, big sets to big numbers). This study is essentially the study of the first step of category theory -- the study of structure preserving maps among similar (via morphisms) or dissimilar (via functors) objects. However, that's not how this study is normally framed -- normally it is discussed as a way to tease out information about the internal structure of the original object. For instance, the dual of homology (the study of weighted sums of points, lines, areas, et c.) is cohomology (the study of functions from points, from lines, or from areas, et c. to a fixed target, typically a field). Whereas (to borrow phrasing from the first article linked in this paragraph) the study of the primal, homology, is intuitively direct, the study of the dual, cohomology, reveals more about the internal structure of the object. Another primal/dual pair appears in Galois theory (and I pick this example not because of a shortage of examples but because it is relatively accessible) between the graph of field extensions associated with (the splitting field of) a polynomial and the graph of inclusions of automorphisms of those fields holding the immediate subfields fixed. This is an example where in one structure, the objects are increasing (field extensions are generically from smaller to larger fields) and the duals are decreasing (fewer automorphisms are possible if a large base field is fixed rather than a small base field). This yields a contravariant functor between the category of field extensions and the category of "groups of automorphisms of extension fields holding their base fields fixed". But it's this latter category that tells us whether a particular polynomial is solvable in radicals. So its the study of the dual, the collection of "what this object does under structure preserving maps" that informs us about the object, not "what this object is".
This latter idea flows from Noether's theorem about duality between symmetries of an object (what structure preserving things it does) and invariants of the object (what the object is) and has informed huge swathes of 20 century physics and chemistry. Conservation of linear momentum (something intrinsic to a system, so "what it is") corresponds with translation invariance (something done by/to the system). Running with this idea suggests studying the symmetries of an object -- the set of structure preserving functions from the object to itself, and then to hunt for patterns in such collections of symmetries. This then flows into the two ideas I wrote about above.
It is a fortunate fact that, while primals do not always answer all the questions we ask, there is usually a dual that can fill in some or all of the gaps. Thus, knowing what an object is is not enough -- we must also know what it does.