That's right. Threads share everything but their stack.

> Does this mean that there is a missed optimization on Linux to preserve some virtually-addressed caches and tables when switching to thread/process that shares the same address space?

I'm not entirely sure I understand your question. What scenario do you have in mind?

Yes the Linux scheduler doesn't differentiate between threads and processes(see NPTL vs LWP) However I was pointing out that although the kernels just schedules "tasks" instead of threads and processes it is far more efficient to schedule threads than processes on some processors - example pre-ARMv6 didn't use address space tags on its TLB so a context switch to a different process required the whole whole TLB to be flushed which means the kernel will need to walk the page table to get the translations for the physical pages for the newly scheduled process.