This is what I'm saying: the stories in the same sandbox share the same world building; or, as you put it, the "entire fictional world."
Okay, but:
Same world = same visuals
Generally, no. Same world = same PROGRESSION OF EVENTS. It doesn't matter as much if the characters or settings
look the same across multiple stories, what matters is that characters in the next story reference events in the previous one as if it is part of their past.
indeed,you contradicted yourself here. Before you conceded that stories in the same world should NOT be inconsistent.
No, I've said many times that stories in the same world can and usually ARE inconsistent with each other in terms of visuals, due to the practical limits of production design (recasting actors being the most obvious change). The story only needs to be consistent with ITSELF.
And lest you forget, what we're talking about in terms of "need" is for the bare minimum of integrity to make a believable narrative. Some properties can get away with a truly hilarious amount of inconsistency because the random and unexpected changes to the visuals are actually built right in to the narrative structure (Doctor Who's famous solution to the Recasting problem). That's yet another datapoint that visual consistency is entirely optional, only NARRATIVE consistency really matters.
No, it doesn't depend. I am talking about related stories.
Yes, so am I. It depends on how closely they're related.
Back to the Future II, for example, is a story in which half the events of the movie take place concurrent with its previous installment. So if you change the visuals for the 1955 scenes, the differences would be a lot more noticeable. A similar thing happens in Terminator Genisys, where the producers go out of their way to carefully recreate the setting, the look and even the atmosphere of the original "Terminator" film.
OTOH, Star Trek gave us an episode where part of the story was actually supposed to have taken place BETWEEN SCENES of a previous story, which also involved an attempt to faithfully reproduce the sets and props of the previous story...
... not entirely successfully.
The names of the production company or the investors are not relevant to the nature of the stories.
That's pretty much the only thing that IS relevant. The people making the show are the people who control what the show looks like and how the story unfolds. If they do a terrible job of telling the story, it doesn't matter how consistent the visuals are (again, These Are The Voyages is a good example of this). If they tell a very good and engaging story, the visuals matter even less.