So is this as impossible as I think or what?
You need to give yourself more credit, it is as you point out deserving of an 'N' followed by a lot of 'o's. A fair few. Noooooooooooope, kind of thing.
I mean, planets are planets and stars are stars am I right?
On the whole, but it's fuzzier than you think. There are stars known as 'failed stars', or Brown dwarfs, that aren't 'star'-ey enough to become complete stars as you'd imagine them, such that they end up more like a gas giant, like Jupiter. There are even a few stars that failed to become failed stars, if you can picture that, and yet are still classified as stars rather than planets.
They don't shift
If they're going to shift, it'd be when they form, so in the case of our Solar System a good 4 or 5 billion years ago. Shifting isn't quite the right word though, an object doesn't decide to start as one then become another, either it has all the mass for a star and the laws of physics smooshes it all up into a star, or it has nowhere near enough mass and it ends up as a planet.