What is(are) the reason(s) for defining things in the following way?

Not only in modern mathematics, but in the modern world. Actually, it always has been this way, but it is more clear now.

What defines your money? Is it the fact that it is printed in paper? Why is it that when you have $2+2+1$ dollars it is the same as if you had a single bill of $5$ dollars? Those are different objects, and different quantities. But both buy the same things.

What happens is that both are equivalent, in some sense. If you know the term and stop to think a bit, equivalence classes are around us in everything we do. It is what makes communication and things practical. If someone writes the set $\{1\}$ in a black board, and someone comes and write the "same set" but with an ugly $1$ they are talking about the same thing. But those things are, strictly speaking, different. They are even in different places on the board.

Talking like this makes it sound pedantry and even useless. And it is. That is what mathematics has understood quite well by now. That such philosophical questions are not quite useful (at least in mathematics), and would lead us astray unnecessarily.

What something is is most times irrelevant, and even difficult to properly define (also in natural sciences, not only in mathematics). However, in mathematics, we have some power over defining things, and this allows us to think that we are saying what something is. But most of times when we are defining things we are actually imposing some artificiality corresponding to what we want those things to do. Take a look at the construction of the integers, rational numbers, complex numbers, tensor product etc. Those constructions are, at the end of the day, just a guarantee that something with those properties exist... which is not to say that it is always easy, on the contrary. For example, the construction of ordinal numbers/cardinal numbers is (at least to me) quite enlightening and elaborate. But they are made with something in mind, and with properties that we want in mind. They are, in some sense, models of some idealization of how those things should behave, not what they should be. Pinpointing exactly how something should behave and how to translate it is one of the most important parts of Mathematics.

What matters is that the $1$ I draw on the board plus the $1$ I draw on the board is the same as the ugly $1$ my friend drew plus another ugly $1$ he can draw. None of them is less $1$ than the other, and to worry about it is literally waste of time.

Short answer: Pragmatism.

It is convenient to use and develop mathematics (especially in engineering disciplines) to maximise its efficiency as a tool.

An example can be found in Wittgenstein's "Bemerkungen über die Grundlagen der Mathematik". It is about why we learn to add in a mechanical way at school. With learning algorithms and defining properties, mathematics will be very transparent for people learning it. There is no obscure idea needed, for instance why we consider something like a determinant and think about its existence. It is helpful at first due to its properties.

In the book, there is an example about addition and why addition must be performed mechanically. In our daily lifes, we do not want to argue how to perform mathematical operations. For instance, when receiving a bill to pay, I am able to add up the prices mechanically and if I find mistakes, I can refer to that algorithm I learnt at school and everyone will agree. This approach excludes any discussion about what numbers are. There is no discussion about the existence of such a function called "addition" needed. The only thing that matters is that they are helpful (which is handwavingly defined without thinking about it much).

This approach justifies why this is important what they do.

I think I cannot give any elaborated answer to your second question, just a quick idea: Assume your approach of mathematics is based on philosophy first, which means that the way you deal with mathematics follows certain philosophical principles. This principles (like: only accepting constructed things, being clear about the existence of mathematical objects, no Axiom of Choice) might be violated if you get some results by an approach using "helpfulness" as a criterion.

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.

They are the consequence of formalism, what is the core of the modern mathematics and it giant developing from principles of XX century.

The point is that it is irrelevant to ask about "what they are" because the answer is "nothing" or "anything".

For science exists a maxima that says: the form is the function, i.e. there is no difference between what they "are" and what they "do".

Notice that the question about "what they are" is metaphysical, variable and subjective.

By the other hand the pragmatism is the core of any science and development, i.e. pragmatism is the requirement for objectivity, from the older Rome empire to today: you know what something is seeing what it does, what are their consequences.

The history of development of humanity is the history to transform "things" to "no-things". The reason is that when something "is" then it is restricted to what it is... but when something "is not" it is free to be used as you want.

One example of this transformation was the change of written languages based on symbols (by example ideograms or hieroglyphs) to languages based on signs (letters, syllables), with the transcendental apparition of the alphabet of the ancient Greek.