There shouldn't be any visible difference if TV respects the signalling.
The setting will change both the (limited/full) range of the pixels output, and the signalling of limited/full range to the TV.
So if the TV understands the signalling (it should) then it should display very similar pictures in both cases.
The setting is only useful when the TV fails to respect the signalling, and e.g. if interpreting the pixels as full range when they are actually limited.
Forcing the pixel encoding will solve that, but it is rarely required (only when TV is causing the problem).
So, I don't think this is the issue.
The TV may have different presets for colour/contrast/brightness from different sources. You may want to experiment in the TV menu.
Thanks and sorry for the late reply.
I haven't been able to achieve satisfactory results with the TV settings and as far as I can see, this seems to be a problem with the Pi. My Pi and my Windows laptop produce very different images with the exact same TV settings.
Here's how it looks like on my Windows laptop (normal colors). Sorry for the poor quality of the photo.
And here's how it looks like on my Pi (oversaturated colors):
The issue persists in OSMC as well, so this seems to be a problem with Kodi and my TV. It also persists over different video files.
Here's a log from Kodi: DiJY
I remember experiencing this kind of problem with a Mac Mini many years ago, and I fixed it after running some commands on the Terminal. I absolutely cannot remember right now what exactly I did to fix the problem.