It is inelegant and invites confusion.
"Can I have a Coke Coke?"
"Do you have Diet Coke?"
"No but we have Diet Dr. Pepper."
"Can I have a Coke (or Pepsi, or any other type of soda)"
"Do you have any diet pop?"
"Yeah, we've got x/y/z"
Just because you can finagle a workaround to use needlessly confusing terms doesn't make them correct.
You don't do this shit with other products.
"What kind of M&Ms do y'all have?"
"Reese's Pieces, and Junior Mints"
"Junior Mints please"
"What kind of Tacos do y'all have?"
"Enchiladas and burritos"
"A burrito please"
"What kind of basketball do y'all play"
"Hockey and baseball"
"Let's go play baseball"
There are terms meant to encompass members of a group. Using the first or most popular member to represent the entire group is a poor choice.