Kernel Trick and Inner Product Preservation
It doesn't mean that $\langle x, y \rangle = \langle \phi(x), \phi(y)\rangle$.

Kernel method in general means that for an algorithm that involve $\langle x, y\rangle$, we can replace it with a function $K(x,y)=\langle \phi(x), \phi(y)\rangle$ where computing $K(x,y)$ is easy. It is known that for $K$ that satisfies Mercer's theorem, there is a corresponding $\phi$ which is the map to the higher dimensional space.

We do not need to know $\phi$ explicitly and $\phi$ can be very complicated.