My only trip to the Thunderbird was summer 1973. I thought it strange at the time that just about ever consumable in the Thunderbird - matches, sugar packets, sugar cubes, silver ware, dinnerware, napkins, glasses ashtrays, swizzle sticks, items in gift shop, etc. were all marked Thunderbird. But when you think about it the hotel / casino needed these items so why not use them for advertisement.
Getting to your question about the syrup, I am sure they found a syrup company that would use their standard syrup in bottles labeled Thunderbird.