Funny, but I initially thought you were describing the "Do everything my way or else you're persecuting me" philosophy evinced by nolongerangry and others in the sex ed thread!
I hear you on this question and I'm on the fence. There are obvious traits that mormons have in common from a cultural perspective. From a physical perspective mormons have a long, long way to go to develop genetic traits that make them more unique but from a cultural perspective I think they are close.
That said, the church is changing so much it may make it more difficult for mormons to maintain their cultural unique-ness. Think about it - no more roadshows, or giant dance numbers (at least none that I'm aware of). The activities are low-to-no-budget so there is nothing special there.
I feel that from my perspective the things that made mormonism desirable have slipped away. Like awesome youth activities and themed dances with crazy decorations (not just a few balloons in the basketball net). It seems the epic nature or scale of the activities has dried up. Now it is just a low-budget group that gives a lot of money and gets little in return.
I would say they should be a cultural identity not ethnicity. Ethnicity has to do with language and color and culture. And Mormons are supposedly color blind, speak different languages, and have different cultures. But if Mormons openly admit that Utah has superior culture, openly admit that they prefer Anglo Saxons, and that English is the most important language, then yes they are an ethnicity.
Which come to think of it they do quietly espouse those beliefs.