C1 uses the "miles" term because many of the consumers they're chasing are the same types of consumers who don't understand airline miles and feel they've been burned by the system (no low level availability, only one airline to use them on, etc). That's why they call their points "miles" with no blackout dates (though there's a cap to their value).
I tried looking into the Mileage Plus Explorer card by Chase because I fly United a lot, but, to my extreme annoyance, I was unable to determine from their website whether you are getting real miles or fake miles.
Airline-branded credit cards will earn the same types of miles as you do when you fly, so the MileagePlus Explorer Card would earn "real" United miles.