What does it mean to "pay X on the dollar"?

It means that you are paying a price of 'X' for every dollar it is worth.

For example, if you are buying a refrigerator worth $100 for $60, you are paying at 60 cents on a dollar.

This expression can be used when something is sold at a premium too. Like, someone can sell you a stock at $1.50 on a dollar.

The use of this phrase can help people calculate profit or loss percentage pretty fast. I guess that could be the origin of the expression too.

To get at the origin part of your question, I've got no definitive answer but was able to find it in print back to at least 1797. Here's an example from early New York State Law authorizing the state to collect a tax on its citizens:

Interestingly, this was not too long after the term dollar was officially adopted as a name for U.S. currency (and only three years after they were first minted) according to this bit on the word from Etymonline:

English colonists in America used the word [dollar] in reference to Spanish pieces of eight. Continental Congress July 6, 1785, adopted dollar when it set up U.S. currency, on suggestion of Gouverneur Morris and Thomas Jefferson, because the term was widely known but not British. But none were circulated until 1794.

I suspect that the origin is in insolvency where debtors only got back part of their claims, measured in X cents on the dollar (rather than X percent), and that this later became a more general phrase for a discounted payment.

Seems like it is used as a "percentage" (%).

"90 cents on a dollar" simply means 90%.

1.5 on a dollar means 150%.

To me it is more confusing. To my ears, % sounds easier.

American expression or Anglo sphere expression?