To start with the cata part of the word, the preposition κατα covers over a page of the minutely printed Lydell and Scott Greek Lexicon. Its complications will not help us much. In brief, though, the general notion is down, albeit confusingly both down as in from and down as in to/at. This is because the genitive case can indicate either origin or target. So kataglōttizō (καταγλωττιζω) meant either that I bad mouth someone (literally 'tonguing down/at') or that I engage in oro-lingual kissing). This is ancient not (as far as I know) modern Greek usage. So ancient Greek language usage developed as careless of strict consistency as any contemporary language has.
Looking at Lydell and Scott, katēgoreō does involve accusation, in the sense of naming before the relevant authority in, yes, the market place [agora]. The verb's object (the accused) has the genitive case, as all targets, whether of spears or of accusations, do. So Plato in the Euthyphro 2c tells his young friend that
It looks as if he <i.e. the accuser> is off to tell him < <katēgorēsōn mou (κατηγορήσων μου-genitive)> to the city as if to his mother. <note the ironic simile with a child running off to his mother to tell on another child>.
So the noun catēgoria is something which any member of a classification can be 'accused' or, in other words, called out/named. A theory of categories in essence a theory of naming. So what can we 'name/accuse' of being a dwelling: a house, bungalow, apartment, tent, yurt, igloo, set, sty, nest, old people's home, mansion, palace, hovel, .., prison,.? Well, you might quarrel with the animal 'dwellings' and whether a prison or a prison cell or the governor's residence are really dwellings. But what would be the point of a theory of categories without our being able to argue about them? In fact a central feature of Aristotle's contribution to more scientific way of thinking was his suggestion that we should define things by means of classification: We define 'dwelling' by the names of the things that are dwellings.
And what is predication if not about naming and classification. Turning to the Latin root for 'predicate', the word praedico - praedicā
re, the first meaning, according to Lewis and Short's Latin dictionary is:
to cry in public, make known by crying in public, to publish, proclaim.
More generally it refers to affirming something to be true. So it is not surprising that grammar acquired the distinction between a subject (what is being talked/written about) and a predicate (what is said to be true about the subject). So the Cambridge dictionary defines it:-
in grammar, the part of a sentence that contains the verb and gives information about the subject:
This is pretty close to katēgoreō and, closer, was more generally used to mean making something known in public.
So there is a semantic connection, even if a subtle one.
In addition, you have to remember that ancient Greek had its own phonetic reasons for its morphology as does any language. In the Attic dialect of ancient Athens the final letter alpha (α), of kata, when run into the first letter alpha of of the next word, agora (αγορα), gets lengthened into eeta (η) from kata-agoreuō/κατα-αγορευōειν to katēgoreō/κατηγορεω.
So there is a link between the market place (as a talking place both for conversation and the conversations/talk/naming of government, legislation and legal prosecution. There is also a connection between the accusatory 'naming' of a person to the 'king archon' on a stated charge and the methodology of Plato's brilliant pupil, Aristotle, in establishing the naming of things as a key part of science.