What is the intuition behind the uniqueness constraint for morphisms determining products? I am following Bartosz Milewski's book and weblectures and have a question about the definition of the categorical product.

He uses names that are a bit different from the ones I can find on Google, so here is the definition. An object $c$ is the product of objects $a$ and $b$ (just denoted $c$, not $a \times b$) if it has the projections $p : c \rightarrow a$ and $q : c \rightarrow b$ such that for any object $c'$ together with projections $p' : c' \rightarrow a$ and $q' : c' \rightarrow b$ there exists an unique morphism $m : c' \rightarrow c$ that make the triangle commute, i.e. such that $p \cdot m = p'$ and $q \cdot m = q'$, i.e. "there exists only one of them that makes the triangles commute". 

Now, what is the intuition behind this uniqueness constraint? I'm not looking for a counterexample, but for the "reason" behind it. What happens when we drop the uniqueness constraint? I know that if we do we satisfy all requirements for $a = a, b = b, c = a \times a \times b, c' = a \times b$, which we should not be able to. When we drop the uniqueness constraint, we allow for the situation that the morphism $m$ "makes up" something by selecting some random value for the unused $a$.

Can we state something about this in terms of bounds or entropy? I.e. something like 'a product is the "lowest-information situation" possible while still having all information available to be general enough to always satisfy the laws' or something like that? Like some greatest lower bound? An by enforcing $m$ to be unique we ensure that it cannot "add entropy/information" in any way?

Edit: I guess what I'm asking is: what is the consequence of one object having an unique mapping to another? If a morphism $a \rightarrow b$ is unique, what does this tell us about the information content of $a$ and $b$? Surely $a$ is "bigger"/contains more information than $b$, right? Doesn't $b$ even have to be a singleton in this sitution?

Edit2: OK let me reformulate my question. When we look at the object that forms the product, we are looking at the /smallest possible object/ that still satisfies all laws. Any bigger object we can "shave down" to this most basic object, and still retrieve the two components. Dually, when defining a sum, we want the most general object possible, any instantiation we can generalize. We define these greatest lower bounds and least upper bounds trough saying there must be an unique morphism between them. My question: how does the uniqueness of a morphism say anything about the size of the objects at its beginning and end?
When you define the product (not a product) of $X$ and $Y$, you are in fact defining 4 things at the same time:


$X \times Y$
$\pi_1 : X \times Y \; \to \; X$
$\pi_2 : X \times Y \; \to \; Y$
$\langle f, g \rangle$, for any $Z$, $f : Z \to X$ and $g : Z \to Y$


Those things must be unique or else this wouldn't be a definition for the product.

The unicity for the second, third and fourth things is expressed with equality. While the unicity for the first thing is expressed with isomorphism, because in category theory we are only interested in objects up to isomorphism.
I thought about it some more, and have been able to come up with an explination such that it makes sense for me.

When we relate morphisms to set theory, if we have an unique $f :: a \to b$, it means $f$ is surjective: it uses the entire codomain. As functions cannot map one element in the domain to multiple elements in the codomain, it means that $a$ is at least as big as $b$. Now, if we require no further conditions, we eventually morph into the smallest set satisfying no conditions, i.e. the empty set. However, if we set some requirements, we eventually morph into the smallest set still satisfying the conditions. This is how a product is defined: the smallest set such that you can construct both parts again, i.e. their union without any extra information. Even more specific: all elements of $a$ and $b$ must be in $a \times b$ as otherwise the conditions will not hold. Furthermore, no additional elements can be inside $a \times b$ because otherwise there would be an unique mapping from it to the set that does not contain this additional element, i.e. it would not be the smallest.

The dual applies for the sum: here we are looking for the biggest set such that it still satisfies the conditions.