Donna Alward pointed out in the comments to the author and branding post that it is more than just a slogan or position statement. And I would agree -- it is a promise to the reader. Just as each Harlequin Mills & Boon series makes a promise to the reader. Within a series, you will find the position statement of the authors are probably closer to each other than authors in a different series.
Every time, a reader picks up one of my books, they should get a passionate romance with accurate history or probably a better way of putting it is a vivid world and an emotionally satisfying ending.
If I changed to writing contemporary, the reader would still hopefully get the same sort of read. So I doubt there would be any need to fashion a new pen name. But I would have to change my position statement.
If I suddenly decided to write guys with gear who go novels or bloodthirsty thrillers with no romance, in many ways, I would be better adopting a new pen name as the bulk of my readers would not be getting what they expected.
Eventually if you become big enough, people do make the connection. For example JD Robb and Nora Roberts. Sometimes though the adoption of multiple pen names happens to quickly. For example Elizabeth Peters writes romantic suspense or mystery and Barbara Michaels writes romantic suspense or mystery albeit with a Gothic twist. They are the same person and probably have a huge crossover audience. I am not quite of the logic behind that one but multiple pen names used to be the norm. And sometimes if you change publishers they want a new name for example Ammanda McCabe is Laurel McKee when she is writing for Grand Central.
Anyway, as an author, you should always be aware of what the bulk of your readership is going to expect from you when you are writing under a specific name. You may want to be creative, but you also want the readers to understand your promise.
I am currently doing my AAs for The Viking's Captive Princess and I must say that I do love my editor. It is reading very smoothly (thus far). But it does mean I have to concentrate.