You are quite right to ask for the context of these definitions. One place to look in this case is
There are three reasons for abstraction:
To cover many known examples.
To simplify proofs by giving the key reasons why something is true.
To be available for new examples.
Thus the power of abstraction is also to allow for analogies.
One should also mention the amazing extension of the notion of metric space by F.W. Lawvere in
Lawvere, F. William Metric spaces, generalized logic, and closed categories With an author commentary: Enriched categories in the logic of geometry and analysis. Repr. Theory Appl. Categ. No. 1 (2002), 1–37.
Later: You should also realise that one of the driving forces of abstraction is laziness! Thus suppose we are working in the space $\mathbb R^3$ with the usual Euclidean distance, and have two points, say $P=(x,y,z), Q=(u,v,w)$. After a time we might get fed up with writing down the formula for the distance from $P$ to $Q$ and decide to abbreviate it to $d(P,Q)$. Then you start asking yourself what properties of $d(P,Q)$ am I really using, and it may be a surprise to find how few of these you need for the proofs, and how much easier it is to use these properties to write down the proofs and to understand them. Thus these properties of $d$ become the underlying structure for this situation. You find that you really understand why something is true. Then you find that these properties apply to more examples, and you are well away to an "abstract" theory.
Again the pressure might be to apply arguments you have used in one situation in another, but the notion of distance does not immediately apply. Hence the notion of "neighbouhood".
After many years it was found that in many situations the notion of "open set" is easier to work with, and has a logically simpler set of rules. So this comes to be thought of as THE definition of a topological space, and the poor students often get presented with this definition with no history, no motivation, no background, but a command to learn it! (Protests not allowed, either! We all know that the writer of:"Give pepper to your little boy/And beat him when he sneezes./He only does it to annoy,/ and can stop whenere he pleases." was a mathematician!)
One of the reasons for abstraction is also that analogies are not between things but between the relations between things. So knots are quite unlike numbers, but the rules for the addition of knots are analogous to the rules for multiplication of numbers. So one can define a "prime knot", and ask: are there infinitely many prime knots? This is how mathematics advances, often for lack of a simple idea. As Grothendieck wrote: "Mathematics was held up for thousands of years for lack of the concept of cipher [zero], and nobody was around to take such a childish step."
Grothendieck has also argued in Section 5 of his famous "Esquisses d'un programme" (1984) against the concept of topological space, as being inadequate to express geometry, or at least the geometry he had in mind. So there is nothing sacrosanct about these concepts, and their applicability and disadvantages need to borne in mind.
In a college debate (years ago!) I was taken to task by a more experienced debater who quoted: "Text without context is merely pretext." I believe that the import of this applies also to mathematics, and relates to my initial remarks.