While I think the "coke" umbrella might be confusing for me personally, I'm sure those raised in "coke" territory do just fine. Like someone else said, it's no different than Band-Aid or Scotch tape, just a brand name so ubiquitous that it becomes a generic term. Lest we forget, "aspirin" was once a brand name of pain reliever, and is now a real English word for acetylsalicylic acid.
I used to work at Blockbuster, and I got my share of people trying to use a different card too, but I think that's a different thing. That's just brand confusion, while the "coke" thing is a colloquialism. I don't think anyone really thinks all sodas are made by Coca-Cola, which would be the parallel to the video store example.