I think you are confusing the TLB and processor data caches.

PCIDs exist for virtual addresses and are implemented in the TLB, not the processor caches. The processor caches are physically indexed, not virtually indexed.

So, does the processor cache have a mechanism to tell which "process" to which a given line belongs? Nope. Two processes (or a process and the kernel) that share memory can be in two PCIDs, but share the processor caches.