Must admit I'm enjoying the discussion over the word "posh" because I'm always interested how language differs between countries and even regions within countries. I think a lot of Australians would understand what you meant by posh, mainly because we share a lot of common language with the UK. Here it does mean fancy but it's also more than that - it means refined, cultured, maybe even "upper class". If you went to a "posh restaurant" you'd mean a very fancy, expensive restaurant. So your "posh voice" is when you try to speak with more refinement, definitely avoiding slang and maybe even using a different accent.
I have no idea about studies but I agree it does seem to be a very common thing. Maybe it's because when we speak on the phone to someone we don't know we are trying to sound "better" and give a good impression of ourselves?
I've also noticed that a lot of people use a very different voice when they are leaving a voice message. But I think that is because they feel uncomfortable doing it!