Each process's virtual address space comprises of user space and kernel space. As pointed out by many articles, the kernel space of all processes is mapped to same physical address in memory i.e. there is only one kernel in the physical memory. But each process has its own kernel stack which is a part of the kernel space. How does same mapping work for all processes with different kernel stacks?
Note: This is the OS agnostic answer. Details do vary slightly with OS in question (e.g. Darwin and continuations..), and possibly with architectural (ARMv8, x86, etc) implementations.
When a process performs a system call, the user mode state (registers) is saved, including the user mode stack pointer. At that point, a kernel mode stack pointer is loaded, which is usually maintained somewhere in the thread control block.
You are correct in saying that there is only one kernel space. What follows is, that (in theory) one thread in kernel space could easily see and/or tamper with any others in kernel space (just like same process threads can "see" each other in user space) This, however, is (almost always) in theory only, since the kernel code presumably respects memory boundaries (as is assumed user mode does, with thread local storage, etc). That said, "almost always", because if the kernel code can be exploited, then all of kernel memory will be laid bare to the exploiter, and potentially read and/or compromised.