You’re not quite envisioning the setup right. These machines use buckets/barrels of ink at a time. You pour the ink in.

And you don’t have one for each Pantone, but you can have a few tanks that you basically set up per run. So in the example where the guy is talking DHL, you might load it with black, pms 2035c and pms 116c. (I assume those are the official colors). When you’re done printing out the million envelops, you dump out any extra ink, and clean the Pantone ink out of the press, and load it with the correct colors for the next job.