To really explain MMX and its successors, you need to go much further back in time. Remember the AN/FSQ-7, the computer used in the SAGE air defense system, with a front panel with many flashing lights that ended up on many science-fiction TV shows as a prop?
Instead of having a normal arithmetic unit that worked on one number at a time, it worked on two numbers at once, so as to be faster in processing vectors representing the geographical locations of aircraft.
Computers developed as possible replacements for that computer, like the TX-0 and the AN/FSQ-32, had accumulators which could do arithmetic on either one long number or two smaller numbers at once, with other choices in some cases. This was very similar to what MMX offered - but with a 36-bit or 48-bit accumulator instead of with 64-bit registers.
This is the antecedent to MMX, unlike the very different form of vector processing used in the Cray-1 computer. Because the Cray-1 computer was so successful as an improvement on the power of other computer systems of the time, and otherwise neither today's computers nor past mainframes embody significant improvements to the design of the IBM System/360 Model 195, which had both out-of-order execution and cache memory along with hardware floating-point, to me it had seemed that the next logical improvement in computers would be to include features like those of the Cray-1.
But for whatever reason, this has not happened; the last surviving example of that kind of vector processing is NEC's SX-Aurora TSUBASA computer, with a CPU that uses HBM and is inside of what looks like a video card; it has less FLOPS than a GPU accelerator, but it may be possible to benefit from its vector capabilities a larger fraction of the time.
And at the time MMX was first introduced, it was believed to stand for Multimedia Extensions.