I can't imagine Apple or MSFT ever offering something that would truly replace it - anything either does is likely to privilege their own OS over any other, and Dropbox has no reason to do that.
The other reason that services like Dropbox shine, at least for me, is that by providing a cloud-mediated sync solution, you don't have to coordinate a synchronous conversation between the devices in question. That's valuable, especially since it enables easy sync with > 2 devices (which also opens the door to selective sharing).
That always-there aspect is also how Dropbox became the de facto mobile file system for iOS (and Android, maybe? I'm guessing -- I'm an iOS user). It's also the default mechanism for sharing files too big or numerous for email (e.g., Aunt Sally's holiday pictures).
My tl;dr is that while Dropbox's initial value prop might not be earthshaking, it's done what it set out to do very, very well for me since I started using it 8 years ago, and I'm very glad to pay them for their service.