Of course I'm talking about mechanics. What else would I be talking about?
The dynamics, aesthetics, and general purpose of such features, of course. But that's beside the point:
Your argument is completely semantic. Saying Shenmue has random battles because technically the battles in the game "feel" random is a completely semantic argument -- it doesn't take into account how random battles actually work in games, and how a single mechanic like this can completely change the feel of a game.
It's not a semantic argument, though.
The extra fights in Shenmue serve exactly the same role as random battles; they are fights that appear, with no effect or importance to the story, in order to provide more action and/or challenge when going from A to B. In some cases they make your character stronger too.
As a designer you wouldn't add traditional random battles in Shenmue III because you already have Shenmue-style "random" battles in the Shenmue series; as such, those worrying about having traditional random battles in Shenmue III have unfounded concerns.
Just like you shouldn't expect task lists and waypoints to appear or a status screen after every battle with XP gains shown; it doesn't mean that Shenmue doesn't have sidequests or levelling - it means that Shenmue handles these things differently, and because it already handles these things it's silly to worry about seeing task lists, waypoints, and status screens appearing.
Of course eating to stay healthy and gaining experience in RPGs is based in reality...I know how life works. What I'm saying is, the way in which Shenmue deals with these things is very different to how they're handled in JRPGs.
But that's what I'm saying too. Why are we arguing?
And no, I'd never argue that Shenmue doesn't have combat just because numbers don't fly out -- that makes no sense whatsoever.
But that's how it's typically handled in JRPGs... or is it that, in fact, it's not about the exact mechanics, and the implementation of something can change from game to game without it being a completely different thing?
TL;DR: The Enoki fights (among a few others) are Shenmue's version of random battles: they're optional, they add nothing to the story, there is no indication that they will happen before they do (and no reason to specifically go where they are), and merely exist to provide a bit more of the punching stuff and occasionally level up your character. Because Shenmue already has an implementation of random battles, Shenmue III won't add traditional JRPG random encounters, because it'd be silly to have it done in two different ways at the same time.