How about "You've Got Mail"? This also qualifies as a Similar but Different. Nora Ephron's version is a remake of the 1940 movie, "The Shop Around the Corner." It, in turn, was based upon the play "Parfumerie" by Miklós László. I can also think of "The Parent Trap." It's another Similar but Different. Nancy Meyers' more recent take is based on the 1961 film of the same name. Per IMDb, that film was written by director David Swift based on the book "Lottie and Lisa" ("Das Doppelte Lottchen") by Erich Kästner. Kastner, BTW, based his version on a 1936 Deanna Durbin film called "Three Smart Girls". So it looks like Similar But Different has been around for ages.

I almost included You've Got Mail and The Parent Trap, both good examples. Similar But Different being around for ages? Heavens yes. How about this? The Biblical story about Noah and the Ark? That's likely based on the Epic of Gilgamesh.

The classic assumed identity story is Cyrano De Bergerac, which spun off a bunch of work-alikes such as Roxanne and The Truth About Cats And Dogs.

Great example, Pliny. I remember enjoying The Truth About Cats and Dogs. And of course Roxanne is wonderful. Both good examples of 'similar but different' stories that work as unique entertainment.

White Sands IMDB: A small southwestern town sheriff finds a body in the desert with a suitcase and $500,000. He impersonates the man and stumbles into an FBI investigation. http://www.imdb.com/video/screenplay/vi1190986009/

Before I forget, because it was such a brilliant piece of acting: Deadringers. Jeremy Irons playing twins.

If audience being unaware of assumed identity doesn't preclude it from this type, Usual Suspects (1995). So sharp and fun, and surprising. Story offers none of the wish fulfillment that other assumed identity stories do (like Shakespeare in Love, where the assumed identity provided passage to spending time with Shakespeare and love), instead it's all about the surprise when real identity is revealed.