Ivory is unscented. It's just 99.44% soap. The remaining 0.56% is good old air.
Ivory floats because it is full of fat, not just because it is full of air. There is an issue with Ivory, which is that it has too much water in it. Back when Ivory was new, in the late 19th century, what P&G recommended was that you should buy 6 bars and put them, unwrapped, on the rafters in your bathroom. As each bar was used up, you should replace it with a new one. This wasn't a marketing ploy to get you to buy bars at once -- exactly the opposite: by letting the bars dry out, they lasted practically forever, which is why P&G stopped recommending it.Ivory soap is mostly whipped air in a soap base. That is why it floats. So, bottom line: air for sale in a soap base.
Is this soap expensive in the US?
Probably the reason it gets mentioned so much is not because of its quality (which is fine), but its ubiquity. Growing up, our family and everyone I knew used Ivory soap exclusively because it was cheap, wasn't heavily perfumed, and it worked. You mention Ivory soap to any American over 30, and there's a 95% chance he or she will know exactly what it smells like. That's why it's used as a basis for comparison as often as it is.