It's true that Caltrain now contracts with Amtrak to run the trains, and has for quite a while. But it's not part of Amtrak and was not part of the original takeover of passenger service (except possibly for the bit south of Santa Clara where the Daylight ran). At first, when it was taken over from SP, operation was probably contracted back to SP with other arrangements made later.

I did ride the "Daylight" a couple of times after Amtrak took over - mostly SF-Oakland/SLO. First time, it was pure SP including the vending machine car; other times, it was more Amtrakish with a cartoon arrangement of cars from various railroads still in their original colors and markings (some with Amtrak "patches" as time went on) but with real food (or at least snack/lounge) service. Once, I was placed in a coach (not SP) that had wheel-powered AC so it only worked when we were moving - got a little warm while sitting for 20 minutes in Salinas and 10 minutes in Paso.