Why are there so many universal properties in math?

I don't really understand why there are so many universal properties in math or why they all need to be highlighted.

For example, I'm studying some Algebra right now. I have found three universal properties that are all basically saying the same thing, although the details are different:

Universal property 1: If $R, S$ are rings and $\theta: R \to S$ is a ring map, then for each $s \in S$, there is a unique map $\hat{\theta_{s}} : R[x] \to S$ such that if $i: R \to R[x]$ is the inclusion map, we get $\theta = \hat{\theta_{s}} \circ i$.

Universal property 2: If $D$ is an integral domain and $F$ is a field with $\phi : D \to F$ a one-to-one ring map, then there is a unique map $\hat{\phi} : Q(D) \to F$ such that $\hat{\phi} \circ \pi = \phi$, where $\pi : D \to Q(D)$ sends $a$ to $\frac{a}{1}$ ($Q(D)$ the fractional field of $D$).

Universal property two was used to prove that in a field of characteristic $0$, the rationals are a subfield, and in a field of characteristic $p$ ($p$ prime), $\mathbb{Z}_{p}$ is a subfield.

Universal property 3: If $R, S$ are rings, $\phi: R \to S$ is a ring map, and $I$ is an ideal such that $I \subseteq \text{ker}(\phi)$, then there is a unique map $\overline{\phi} : R/I \to S$ such that $\phi = \overline{\phi} \circ i$ where $i: R \to R/I$ maps $a$ to $\overline{a}$.

It is really hard for me to keep track of all of these universal properties, especially when they are all usually referenced by the single name "universal property". Is there a point to all of these universal properties?

Honestly, I don't even know if my question is clear, or how to ask a better question in this regard.

A universal property of some object $A$ tells you something about the functor $\hom(A,-)$ (or $\hom(-,A)$, but this is just dual). For example, $\hom(R[x],S) \cong |S| \times \hom(R,S)$ is the universal property of the polynomial ring (where $|S|$ denotes the underlying set of $S$). Conversely, we may consider the functor which takes a commutative ring $S$ to $|S| \times \hom(R,S)$ and say that it is a representable functor, represented by $R[x]$. This can be also interpreted as the statement that $R[x]$ is the free commutative $R$-algebra on one generator, see free object for categorical generalizations. Roughly, representing a functor means to give a universal example of, or to classify, the things which the functor describes. This happens all the time in mathematics. Conversely, whenever you have an object $A$, it is interesting to ask what it classifies, i.e. to look at $\hom(A,-)$ and give a more concise description of it. The Yoneda Lemma tells you that all information of $A$ is already encoded in $\hom(A,-)$.

Also, one of the main insights of category theory is that it is very useful to work with morphisms instead of elements. For example, what the quotient ring $R/I$ does for us is not really that we can compute with cosets, but rather that it is the universal solution to the problem to enlarge $R$ somehow to kill (the elements of) $I$. In other words, $\hom(R/I,S) \cong \{f \in \hom(R,S) : f|_I = 0\}$. This makes things like $(R/I)/(J/I) = R/J$ for $I \subseteq J \subseteq R$ really trivial: On the left side, we first kill $I$ and then $J$, which is the same as to kill $J$ directly, which happens on the right hand side. No element calculations are necessary. (On math.stackexchange, I have posted lots of examples for this kind of reasoning.) Quotient rings, quotient vector spaces, quotient spaces etc. are all special cases of colimits.

The universal property of the field of fractions states that $\hom(Q(D),F) \cong \hom(D,F)$, where on the right hand side we mean injective homomorphisms. This says that $Q(-)$ is left adjoint to the forgetful functor from fields to integral domains (in each case with injective homomorphisms as morphisms). This is a special case of localizations. Adjunctions are ubiquitous in modern mathematics. They allow us to "approximate" objects of a category by objects of another category.

So far I have only mentioned some patterns of universal properties, but not answered the actual "philosophical" question "Why are there so many universal properties in math?" in the title. Well first of all, they are useful, as explained above. Also notice that many objects of interest turn out to be quotients of universal objects. For example, every finitely generated $k$-algebra is a quotient of a polynomial algebra $k[x_1,\dotsc,x_n]$. Thus, if we understand this polynomial algebra and its properties, we may gain some information about all finitely generated $k$-algebras. A specific example of this type is Hilbert's Basis Theorem, which implies that finitely generated algebras over fields are noetherian. Perhaps one can say: Universal objects are there because we have invented them in order to study all objects.

Any time $X$ satisfies a universal property, it means that the inventor of $X$ chose well (rather than arbitrarily) how to define $X$.

So I guess the literal answer would be "Because people are telling you about good mathematics".

I had the same question when I was learning algebra from Grillet‘s Algebra book (GTM 242). At that time, I did understand how I could define concepts (i.e. tensor products, free products and free groups, etc.) with universal property (UP), how maps factor through one another, and how the object defined is the “most general of all”. However, I failed to see how UP is important in other contexts (quotients in topology/groups in particular) even though Grillet stated a lot of theorems in UP, as I felt that it is at most a fancy description with commutative diagrams. And yes, I resonate with you when you said "it is really hard for me to keep track of all of these UPs".

However, now that I've learned more, I found that UP is a very useful tool when you are given a map $f: X \rightarrow Y$ and you are trying to find a corresponding map from $\tilde{f}: \tilde{X} \rightarrow Y$, where $\tilde{X}$ is some object you've constructed from $X$ with the map $\phi : X \rightarrow \tilde{X}$. As long as $\tilde{X}$ and $f$ is well behaved (i.e. satisfies the hypothesis for UP), you can get a unique $\tilde{f}$ with similar good properties that $f$ possesses. Even better, you get to factor through the map from $X \rightarrow \tilde{X}$.

This idea might sound cliche, since the proofs of UP usually involve constructions of $\tilde{f}$ induced by $f$ in the most obvious way. But in many more advanced subjects, we are defining the $\tilde{X}$'s (some of which seemed very messy) from intuitive spaces like $X$, while we only know about maps from $X \rightarrow Y$. With universal property, you don't need to explicitly state what $\tilde{f}$ is and simply acknowledge its existence and uniqueness, and this is extremely helpful if you are juggling with both spaces and trying to construct such maps a couple of times in one proof. I realized this when I was learing algebraic topology and proving homotopy equivalence of a space with its quotient space (under certain conditions), and it was then when I suddenly realized the power of the universal property in quotient topology.

A final note on "hard to keep track of all UP's": I don't think you need to actually remember the statements of all UP's, because as long as you know how $\tilde{f}$ is constructed, you could easily recall the UP when you need it. It's okay to find it useless when you encounter statements like this for the first time, but keep in mind that UP will be of much greater significance as you learn more mathematics.