Why is category theory not just another theory? Consider category theory as one theory among many others: with a simple signature and some simple axioms. 
Compare it with - e.g. - group theory as another theory with a simple signature and some simple axioms.
Compare it with set theory as still another theory with a simple signature and some (not so simple) axioms.
How could you tell in advance that (especially and somehow exclusively) category theory gives rise to (and makes definable) such a fundamental concept like universal property? 
Consider the way universal properties are defined: Why isn't one able to define comparable abstract and useful concepts on top of groups, sets, and so on? Or is one?
What makes categories special in this respect - from an abstract point of view?
 A: I am going to rephrase your questions, the way I understand them. I hope this is a reasonably correct interpretation.
1- question: how can you tell by just looking that category theory (CT) (a theory with a simple signature and some simple axioms, in your words) is more powerful than say group theory or.....
Answer: take CT axioms (for ex. Here) and take monoid theory (MT) axioms (for ex. Here). You see almost immediately that MT is a special case of CT where all identities are the same. You also see that groupoid theory (GdT) is a special case of CT when you assume that all arrows are invertible. Since group theory (GT) is a special case of MT or GdT you already see that:
CT is more general (thus presumably more powerful) than MT, GdT or GT. That's not bad, considering that all this can be seen almost immediately.
If you specialize CT is more complicated ways you can prove that the resulting theory is a model for (a flavour of) Set theory. And so on with Rings, topological spaces, ect.
So we can say that the simple axioms of CT can be augmented to obtain many other previously known mathematical structures.
CT - as defined above - has been generalized further with higher categories so I would not say that it is unique in any (permanent) sense. Perhaps in the future we will abstract even more with some other theory, who knows. At the moment Category theorIES are at the forefront of generality and abstraction. That's all we can say.
2- question: Universal properties (UP) are logical statements expressible in the language of CT. Can we guess by looking at their formal structure that they are going to be a fundamental concept in CT?
Answer: I do not think this is possible at the moment. We can feed a computer with a theory and a statement and ask whether the statement is true or not (a Theorem). But we cannot decide - by just looking at the structure of the statement - whether it is going to be very useful or just moderately useful in the future development of the theory. This can only be decided ex-post. In the case of UP, they are not even theorems, just properties (definitions basically), which may or may not apply to a specific category/functor.
They turn out to be fundamental concepts by the fact that they appear to be satisfied by many important categories/functors.  Ex-post unfortunately.
Samuel defined UP in 1948 and Kan went on with adjoints in 1958. CT was founded in 1942. So UP and adjoints were not obvious things.
3- question:Why isn't one able to define comparable abstract and useful concepts on top of groups, sets, and so on?
Answer: even the most abstract construction in group theory is just something that applies to groups (and some derived set or ring, or...) only. It will never be automatically applicable in a non-group setting (topological spaces which are not groups, for example). 
Conclusion.
It seems that much axiomatic mathematics has been developed by "reverse-engineering". You take some nice theorem (Pythagoras' theorem for ex. ) and you work backward to find axioms such that the theorem can be deduced from them. This s apparently what Euclid did. Perhaps UP were invented that way. Basic CT axioms certainly were developed that way.
hth
A: A theory that people actually study is rarely the result a a randomly chosen collection of axioms. If I recall correctly Saunders Mac Lane said categories were defined so that functors could be defined and functors were defined so that natural transformations could be defined. As Asaf Karagila points out transitive sets pop out of set theory. So to universal properties pop out of category theory. Situations like these make for interesting mathematics.
