The stack pointer is just that, a pointer. It points to a region of the heap. It can point anywhere. It's a data structure the assembly knows how to navigate, but it's not some special thing. You can point it anywhere, and change that whenever you want. Just like you can with any other heap-allocated data structure.

It occupies the same L1/L2 cache as any other memory. There's no decreased access times or fetches other than the fact that it just happens to be more consistently in L1 due to access patterns. And this is a very critical aspect of the system, as it also means it page faults like regular memory, allowing the OS to do all sorts of things (grow on demand, various stack protections, etc...)