Most superheroes are, psychologically speaking, more-or-less regular people who adopt a superhero guise to fight crime. Hal Jordan turns into Green Lantern, Peter Parker becomes Spider-Man, and so on. In his modern incarnations, even Clark Kent (who is not precisely "normal", but is as close as he could get under the circumstances) becomes Superman.
Batman is different. The vigilante known as Batman is that person's "default" psyche. He adopts the Bruce Wayne guise in order to function in society in any context other than crimefighting. Even his own comrades don't work this way: Dick Grayson became Robin during his tenure, Barbara Gordon became Batgirl (and then Oracle, and then Batgirl again), and so forth. But Batman turns into Bruce Wayne.
This is a concept that has been explored in a number of Batman-related media. The most stark example may be in an episode of Batman Beyond, where a villain is projecting voices into his head. He realizes that this is a trick when the voices start to speak of him as Bruce, because, as he puts it, "that's not what I call myself in my head." Another example comes from The Sandman, which shows us the dreams of various superheroes. Most of them show up in their civilian guises in their dreams, but Batman does not.
Given all of this, it makes sense that he would talk to himself in the Batvoice: that's his normal head-voice. His "normal" voice is actually the fake one.