He is able to exit Hyperspace between the Starkiller’s shields and the planet’s crust, a feat lifted from an early script of TPM where only Anakin’s Force ability was able to manage such a thing.
And no, the Force has always consistently been something that anyone can learn, but that some have a natural affinity for. That hasn’t changed.
Then why did Rey just suddenly get her powers, without working for anything? It’s explained very clearly in TLJ as the “darkness rises, and light to meet it”. Since Kylo trained for many years to become a dark lord, Rey must get all those powers within a couple days to restore balance (probably because Luke shut himself out).
I’m guessing that Kylo trained for years to become a Jedi, then relatively quickly embraced the ‘easier, more seductive’ pull of the Dark Side to quickly gain more power. The fact that Rey is so easily able to gain such abilities as well would make sense if she were using the Dark Side (and there’s some evidence that she is, such as Luke saying ‘you went right to the dark’), but absent that it seems like she just has more ability than Kylo.