REPL itself and windowing is quite separate things. REPL is just a program, that runs in some textual environment that can print and consume text, hence read eval print loop. So it’s not strictly related to creating a window in Kakoune or Emacs, but creating interactive buffer to hold what was printed, and read what was put.
Interactive buffers in Kakoune are kinda normal thing, e.g. *grep*, or *make*, where you can see the log and jump to errors/grep results. My little flygrep command is another instance of using interactive buffer in Kakoune. Tagbar and Kaktree could be external programs but I see no point, because they were made for Kakoune in the first place and use internal text editing primitives.
So my point is that we can use Kakoune as a rendering engine, just like what Emacs does with it’s buffers. I see big potential in this approach, because it leverages abilities of Kakoune to accept input and render colored text, thus makes it kinda portable. Portability is not an issue though, because of narrow scope.
REPL is something like interactive hacking environment, and I would like to keep all my editor-powers in it, such as completion, structural editing (paredit or parinfer), syntax highlighting, documentation popup:
I mean, that’s really cool to have in REPL. I do not need this for terminal, but I don’t mind having it there as well.
true, but if you’re already in the editor, why would you want this? 
Well we’re on the different sides here, but I think you will agree that if you can have generic repl that works with multiple languages, and has all your personal configurations from the editor you’re using, you will not want to loose it and use external repl.