I'm a U.S. native. I've NEVER heard "to shift on a dime." Never. I've OFTEN heard and said "to turn on a dime." The literal reference is to a car's ability to make very sharp turns. A car that can make a sharp turn can "turn on a dime." "That's a very tight parking space, are you sure you can park there?" "Just watch me!" "Wow, your car turns on a dime, doesn't it?" I've also heard "to stop on a dime." I can only guess as to the origin. A dime is, physically, the smallest U.S. coin, and is sometimes used to mean something tiny.

Extending my answer and responding to a comment... Although I've NEVER heard the expression, or read it in print before seeing this posting, it obviously means the same thing as "turn on a dime." That's clear from River Zedd's examples and others I've found. A Google Books search shows NO hits before 1997, the first in 1997, 21 from 1997 through 2016. (Most or all are U.S. publications). In contrast, "turn on a dime" gets about twenty hits before 1997, about two hundred from 1997 through 2016. So, it is much less common than "turn on a dime," and it is a new phrase that's sprung up in the last twenty years. Obviously it's a new variation on the old phrase. That often happens. I don't believe it is more formal than "turn on a dime." The word "dime" itself is informal. It's clear from reading over the examples Google found that, like "turn on a dime," "shift on a dime" is found in "prose written in a conversational style" (such as the writing of Mark Twain).

It means to change direction or course suddenly. The British equivalent would I guess be 'to turn on a sixpence.'

http://idioms.thefreedictionary.com/turn+on+a+dime See the second meaning. This is a US idiom, and this particular meaning of it is not generally known in the UK. So I learnt something new today too!

I'm curious where you saw this. As Dan said, this is not a U.S. idiom. Dan listed the correct U.S. idioms.