Are cache contents specific to a process?
It depends, there are several kinds of caches.

Some caches, known as PIPT (physically indexed, physically tagged) operate entirely on the physical address. The cache subsystem is independent of virtual memory, so it doesn't matter which process is accessing the cache: all processes share all cache entries.

Some caches, known as VIVT (virtually indexed, virtually tagged) operate entirely on the virtual address. In this system, a cache entry is only valid for a given process. There are several ways to ensure that a cache entry will not be used in the wrong process:


The operating system may flush the cache at each context switch. This requires no hardware support but is obviously inefficient unless context switches are rare.
The operating system may assign disjoint sets of addresses to distinct processes. This again requires no hardware support, but is a heavy burden on the operating system (it means that the MMU is effectively as a mere MPU (memory protection unit)).
The MMU and cache may contain an extra label which is set by the operating system to an identifier for the process that owns each page or cache entry. ARM and Intel call this label ASID (address space ID). Each cache entry is only valid for a virtual address with the matching label.
There are also VIPT caches (virtually indexed, physically tagged). The physical tagging ensures that the cache entry will only be served to a request for the correct chunk of memory. The virtual index means that if a page is shared at different addresses in different processes, the cache entries will not be shared.


See the “Address translation” section in the Wikipedia article for more information on these kinds of caches.

Cache types typically vary between processor families within the same processor architecture. Even on a given processor, data and instructions caches may have different types, as can L1 vs L2 (and more) caches.