I believe that one of the oldest and most important motivations for / settings for categorification hasn't been explicitly mentioned yet (here or in the linked questions). It is due to Grothendieck (and Weil): the relation between cohomology and counts of points for varieties over finite fields. [This happened to be last week's lecture in a class I'm teaching, so a ready-made rant.]
As was explained categorification is understood in relation to DEcategorification, which can be made a well-defined mathematical process: replace a vector space or chain complex by a number (its dimension or Euler characteristic), an associative algebra or category by a vector space / chain complex (its cocenter / trace / Hochschild homology), a monoidal category by an associative algebra / category etc --- these (and many more) are captured by the notion of "dimension" of a dualizable ("finite dimensional") object in a symmetric monoidal [higher] category (which defines an endomorphism of the unit, taking the place of a "scalar").
One can use dimension in this sense as a general definition of decategorification. This has a natural motivation from topological field theory, where taking dimension corresponds to crossing with a circle -- $dim(Z(M))=Z(M\times S^1)$. (This sometimes give a more naive decategorification than passing to K-groups, though in examples the two often coincide or the simpler "dim" is often what we really want.)
But if you have a dualizable object you can talk not only about its dimension (trace of the identity), but about the trace of an endomorphism. For example in TFT you can study not $M\times S^1$ but the mapping torus of a diffeomorphism of $M$ to get the trace of the induced map on $Z(M)$.
In the setting of geometry over finite fields, there's a canonical choice of endomorphism to take, namely Frobenius, so we get a different version of decategorification by consistently studying $Tr(F)$ instead of $dim=Tr(Id)$, and correspondingly a different notion of categorification.
The Grothendieck-Lefschetz trace formula tell us that l-adic cohomology categorifies (in this general sense) counts of points over finite fields (as captured by zeta- and L-functions), a categorification just as revolutionary as the replacement of Euler characteristics by homology groups. But this is just the beginning. Note also that this categorification is richer in the sense that we can take traces of powers of Frobenius to find point counts over all finite extensions of our finite field.
Grothendieck's function-sheaf dictionary is a fundamental categorification, a relative version of the above: it suggests that interesting functions on sets of points of varieties over finite fields can be categorified by l-adic sheaves; or all together, interesting function spaces are categorified by categories of sheaves.
This idea is behind much (most?) of modern geometric representation theory, and in particular one of the most spectacular achievements in math: the entire representation theory of all the finite groups $G(\mathbb F_q)$ for $G$ reductive (like $GL_n, SL_n,SO_n,...$) -- a list that includes almost all the finite simple groups -- was categorified in this sense in the collected works of Lusztig. This includes the Deligne-Lusztig construction of representations, the celebrated [first set of] Kazhdan-Lusztig papers, which can be interpreted as categorifying the [unipotent] principal series representations (closely related to Springer theory categorifying the representation theory of Weyl groups), and the theory of character sheaves, which roughly categorifies the entire character theory of these groups. In other words, a huge amount of what we know about this huge family of finite groups comes from categorification.
..and the entire Geometric Langlands program comes from this idea, starting with Drinfeld, taking the same philosophy from reductive groups over finite fields to reductive groups over local fields. The Geometric Satake correspondence categorifies the classical Satake correspondence, the basic mechanism behind the Langlands program, and is fundamental to our undrestanding of what local Langlands is about (ie how to organize representations of p-adic groups) thanks to Fargues-Scholze. V.Lafforgue's proof of the automorphic-->spectral direction of Langlands for function fields is inspired by the categorification idea, and the work of Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky explicitly makes sense of this process, recovering spaces of [unramified] automorphic functions over function fields as trace of Frobenius on suitable categories of sheaves.