They are generally referred to as fancy shaped diamonds. In fact, they can appear bigger, particularly when set in a halo setting. That is one of the reasons, why people yearn for them.
When Sugarland created the fancy cars and diamond rings lyrics, they must have meant something along the lines. Pop culture somewhat reflects the aspirations and desires of a materialistic society. Being “fancy”, they offer a contrasting shape to what is considered traditional.
Note that the term “fancy” is attached to diamond colors also, which need not be mistaken with their shapes. Most people who get diamonds as a gift or for their engagement have a preferred shape or two in mind. While other factors like diamond color, quality, and price should be determined also, the choice of its shape is something personal.
In other words, if you are buying a diamond ring for someone, you should get to know the+nd shape she prefers the most. This is particularly true for fancy-shaped diamond engagement rings.