Well, after it was mentioned in a quiz TV show, I suddenly discovered that Bruce Wayne owns the Daily Planet, and so I wanted to ask. In any comic has it been shown that Batman knows that Superman's alter-ego works for him, or that Superman knows that Batman's alter-ego Bruce Wayne is his boss? Or do they not know that little detail?
In the Batman: Hush storyline, it is named a subsidiary of Wayne Entertainment.
Batman has known Superman's secret identity for a long time (since Superman Vol1, #76 in 1952) so it's unlikely he wouldn't have known he was Kent's boss. It's even weirder because he bought it from Lex Luthor
Both Wayne and Kent know the truth about each other’s alter egos by the period when Wayne purchased The Daily Planet, so we can guess this was certainly some sort of power move by Batman. Perhaps Wayne’s main motivation for buying the The Daily Planet was to save the paper from Luthor, but let’s be real about these fictional superheroes, there’s no way ego wasn’t a part of his calculation.
How much influence Wayne has had over the publication’s reporting remains unclear. Presumably, Wayne had a few tips on dimly lit, tall building ledges for the crime beat reporter. But unfortunately, “Help!” writer Jeph Loeb was unavailable to clarify the motives of the characters, as he now works as the executive vice president of Marvel Television.