It is pretty straightforward to show that if $X$ is a Banach space under two equivalent norms, then the respective open unit balls $B_1$ and $B_2$ are homeomorphic only by showing that $B_i$ is homeomorphic to $X$ (e.g. via $x\mapsto {\rm tan}(\frac{\pi}{2}\|x\|)x$).
Can we show that the unit spheres (defined in terms of two equivalent norms) are homeomorphic to each other?
I am interested mainly in the finite-dimensional case.
In finite (fixed) dimension all the norms are equivalent. The function $$x\longmapsto\frac{x}{\|x\|_2}$$ is continuous and bijective form $S_1$ (compact by Heine-Borel) to $S_2$, so the inverse is continuous.