I didn't pay that much attention, because, believe it or not, I don't actually own any Caribou travel mugs drinkware (though I do have a couple of regular mugs). While I'm not completely sure how that happened, those are the facts about my travel mug stash.
I was browsing their official Facebook Fan page (which, of course, I've "liked"), and I saw this in the conversations (click on the image to enlarge):
So, it sounds like they're encouraging you to use Caribou mugs (for obvious reasons), but allowing you to use any mug. Further down in the thread, someone pointed out that it's like grocery stores giving you a 5 or 10 cent discount for bringing in a reusable bag of any kind, because that also saves them money. I thought that was a good point.
So, I'm planning on giving it a try, with my Starbucks mug. I do love Caribou mugs, but Starbucks has this line of dishwasher safe (very important) mugs from around the world that I just love.
Getting back to the point, I'll report back with my findings, but my guess is that each store might do things a little differently, so I'd give it a shot no matter what. Good luck saving .50!