Specifying Coke carries an implicit "not Pepsi"

My point is that it doesn't. The name "Coke" has evolved into a generic term. Just like "Kleenex". Your preference for the specific brand, although understandable, doesn't change this fact.

I'm sure that the companies which own those trademarks are very proud of that.

If you're at a friend's house and you've got a runny nose and say, "Can I have a kleenex please", I'm sure you would be surprised to have them respond with, "Sorry, I only have a generic brand of facial tissue. Is that OK?"