SIMD does stand for Single Instruction Multiple Data [0]. So, the hardware performs some linear algebra operation on one or more vectors. Whether the vector is stored in unique registers, combined registers (as described above), or memory buffer(s), that is up to the hardware designer. In fact, on x86, the MMX registers were aliases to the normal floating point registers [1], and whereas on SSE, new registers were added [2].
[0] https://en.wikipedia.org/wiki/SIMD
[1] https://en.wikipedia.org/wiki/MMX_(instruction_set)#Technica...
[2] https://en.wikipedia.org/wiki/Streaming_SIMD_Extensions#Regi...