The order of a torus link consists of a pair of integers $(m,n)$, with at least one of them nonzero, and it is such that if the two integers are not coprime, i.e. of the form $(km, kn)$, the link forms $k$ interlinked copies of the $(m,n)$ knot.
Because of this, I see a clear correspondence between these orders and the rational numbers $\mathbb Q$ (possibly supplemented with a point at infinity, if required?), which also consist of pairs of integers with equivalences between pairs that are multiples of each other.
Is there any nice reference that formalizes this correspondence, or which uses it explicitly? The more 'textbook-y' (which is to say, the more suitable for use as a reference for this fact in a separate context), the better.
A torus link is any link that is isotopic into the standard unknotted torus in $S^3$ (the Clifford torus, if you like). So, the problem of torus link classification can start with the question of what are the possible links in $T^2$ itself, with all the isotopies restricted to the torus.
Take a look at the section "Knots in the torus," starting on page 17 of Rolfsen's Knots and Links. A first step of link classification can be knot classification, i.e., the classification of simple closed curves in $T^2$. There are two main types: separating (equivalently, nullhomotopic or bounds-a-disk) and non-separating. The separating ones are just unknots, so let's exclude them. The non-separating curves are in one-to-one correspondence with non-divisible elements of $H_1(T^2)$, and if you choose a basis for $H_1(T^2)$ you get the pair-of-integers representation $(a,b)$ with $a,b$ coprime. Conceptually, the argument is that in the universal cover, the loop lifts to a path between integer points, and by a homotopy you can pull the line tight and it has rational slope.
For links on a torus, you can use intersection numbers to see that all the components have to be parallel (meaning they all have the same slope). There is a standard construction to represent an arbitrary element of $H_1$ as a collection of simple closed curves, and in $T^2$ this element is unique up to isotopy. For $(ka,kb)\in H_1(T^2)$ with $a,b$ coprime and $k\geq 1$ (which encompasses every non-zero element of the homology group), the construction gives $k$ parallel copies of the $(a,b)$ knot in $T^2$.
Calegari's book on foliations says a little at the beginning about the relation between homotopy and isotopy in a surface: https://math.uchicago.edu/~dannyc/books/foliations/foliations.html