No, you cannot consider them different sides of the same entity.
Fuji is the station. They can dictate what is shown on their channels, and Toei has no say-so or further association with them other than partnership.That's like saying "Well Sony's animation studio made The Spectacular Spiderman show but it aired on Kids WB, so Sony and Warner Bros are the different parts of same entity, head and body".
Which obviously is not true.So yes, Toei makes the decision, but they have to follow requests of sponsors and stations.
And in this case, the station was the one that made the decision, Toei must follow.
So if you're gonna raise pitchforks and torches, head to Fuji not Toei.
I meant to say something along those lines, thanks for the clarification. It's hard to come up with a metaphor for this kind of thing.