In many posts on MSE, it is discussed that Cauchy sequences can't be defined in General topological spaces and in a typical topology book it is discussed what converging sequences are, but, what I don't understand is, why, on an abstract level, does convergence generalize even without a metric while cauchy-ness doesn't?
My issue with the posts (eg) is that they show that Cauchyness is not a topological property using specific sequences. But what I wish to know is the abstract idea behind it.
The first abstract idea to come to terms with is this:
The following statement is false: For every topological space $X$, for any two metrics $d,d'$ that generate the topology on $X$, and for every sequence $(x_n)$ in $X$, the sequence $(x_n)$ is a Cauchy sequence with respect to $d$ if and only if it is a Cauchy sequence with respect to $d'$.
In your post you wrote
"... they show that Cauchyness is not a topological property using specific sequences. But..."
... But ... an important thing to realize is that those specific sequences amount to a proof that the above statement is false. All one needs for that proof is to give one counterexample, i.e. one example of $X,d,d',(x_n)$ such that $d,d'$ generate the topology on $X$, and $(x_n)$ is a Cauchy sequence with respect to $d$, and $(x_n)$ is not a Cauchy sequence with respect to $d'$.
With that out of the way, your question regarding the abstract idea behind "Cauchyness" is still very interesting. The comments give two distinct answers to this question: that abstract idea can be uniform structures; or it can be Cauchy structures. One thing I'm unsure of is whether these two notions of "Cauchyness" are equivalent, but I believe they are, in fact I think that this equivalence is covered in this part of the first link above (if I'm wrong about this, I hope someone corrects me).
Okay, so let me try to answer your question by explaining why a uniform structure captures the abstract idea behind Cauchyness. Consider a sequence $(x_n)$ in a topological space $X$.
Assuming that the topology on $X$ is generated by the metric $d$, let recall the standard definition: to say that $(x_n)$ is a Cauchy sequence means that for any $\epsilon > 0$ there exists an integer $N \ge 1$ such that for any integers $m,n \ge 1$ we have $|x_n-x_m| < \epsilon$.
But now, let me reword this definition in a vague and intuitive fashion, without referring to the metric nor to any number $\epsilon$: Given some uniform measurement of closeness in $X$ there exists an integer $N \ge 1$ such that all of the terms of the sequence starting from $x_N$, namely all of the terms in the set $\{x_N,x_{N+1},x_{N+2},...\}$, are simultaneously close to each other with respect to that given uniform measurement of closeness.
What goes wrong in an ordinary topological space $X$ is that there is no such thing as a "uniform measurement of closeness". At best, an open neighborhood of a point $x \in X$ gives a local notion of closeness around $x$, in the sense that being an element of that neighborhood is a notion of "being close to" $x$. But "closeness to $x$" is not (and should not) be transitive! That local notion gives you no way to test whether all the points in an entire infinite set like $x_N,x_{N+1},x_{N+2},...$ are simultaneously close to each other.
The idea of a uniform structure is that a "uniform measurement of closeness" can be expressed by choosing a subset $U \subset X \times X$. Having made a choice of $U$, one can then declare that the points of a set such as $\{x_N,x_{N+1},x_{N+2},...\}$ are all uniformly close to each other if and only if all ordered pairs drawn from that subset are elements of $U$. A uniform structure is then defined to be a set $\Phi$, each of whose elements $U \in \Phi$ is a subset $U \subset X \times X$, such that $\Phi$ satisfies a bunch of axioms.
Here's an exercise: show that if $d$ is a metric on $X$, and if for each $\epsilon > 0$ we define $U_\epsilon = \{(x,y) \in X \times X \mid d(x,y) < \epsilon\}$, then $\Phi = \{U_\epsilon \mid \epsilon > 0\}$ is a uniform structure on $X$.
One can then define a sequence $(x_n)$ in $X$ to be Cauchy if for every $U \in \Phi$ there exists an integer $N \ge 1$ such that every ordered pair drawn from the set of terms $\{x_N,x_{N+1},x_{N+2},\ldots\}$ is an element of the set $U$.
Here's my attempt to explain.
Consider this definition of convergence that does not require a notion of metric:
A sequence $\{x_n\}_{n\in\mathbb N}$ converges to $x$ if and only if for each open neighborhood $U$ of $x$ there is an $m_U \in \mathbb{N}$ such that $x_n \in U$ whenever $n \geq m_U$.
Now, try to imagine an analogous definition for Cauchy sequences. In the convergence definition, we don't need to shift $U$ in anyway. In the Cauchy case, we would have to say something like:
A sequence $\{x_n\}_{n\in\mathbb N}$ is Cauchy if and only if for each open neighborhood U of the origin, there is an $m_U \in \mathbb{N}$ such that $x_j,x_k \in (U + x_m)$ whenever $j,k \geq m_U$.
Hence, the difference is the Cauchy definition has this implicit "shifting" of open sets, and such a mechanism requires an "origin" and a metric.
