Question: Think about it: capitalism has brought so much prosperity to so many people.

Yet no country claims our national ideology is capitalism - none that I know of.

Americans can proudly claim they are capitalists. Well. First. Not so much. And then, it's not technically the American national ideology.

There are "Socialist states of...". There are "Islamic states of...". Then there is "Democratic republic of..."

What's funny is that countries that claim they are democratic are usually not democratic either. North Korea is called the Democratic People's Republic of Korea. But that would be another question.

Why does no country ever claim capitalism as their state ideology?

Why is there no "Capitalist Republic of America"?

Answer: 
Capitalism as we currently understand it in the modern sense is a political philosophy that requires the government to institute policies to create capitalist markets. For example, there is no way to create a joint-stock company without laws to establish them as legal entities and to regulate them. Read [Debt: The First 5000 Years](https://en.wikipedia.org/wiki/Debt:_The_First_5000_Years) and specifically the part about the "Iron Law of Liberalism" for more details about the need for governments to create Capitalist Markets.

However, due to the fact that since the period of European Capitalist Colonialism (e.g. East India Company was a Joint-Stock Company) the dominant world system has been Capitalism. Since it was the default, there was no need call a country as such. Most socialist countries are called that in response to the default of Capitalism. Same thing with Islam, its not that their culture is more focus on religion but that those stats are established in response to previous secular rules. The Islamic Republic of Iran is called that in response to the [more secular Shah](https://en.wikipedia.org/wiki/Iran#Contemporary_era) that had been previously installed by the US & Britain. Afghanistan under Taliban rule was called ["Islamic Emirate of Afghanistan"](https://en.wikipedia.org/wiki/Islamic_Emirate_of_Afghanistan) but that's after a secular Soviet invasion. Saudi Arabia's formal name on the other hand is "Kingdom of Saudi Arabia" despite the fact that they are very religious because its not response to secular rule.

In a similar vein, when Bolivia changed its constitution in 2009 to give more power to its indigenous communities, it also changes it's formal name to the "Plurinational State of Bolivia" to [recognize its multi-ethnic nature](https://en.wikipedia.org/wiki/Bolivia) in contrast to previous domination by people with Spanish ancestry.
Apart from all the other answers, no nation should call itself "Capitalist Republic of ..." because there isn't currently any nation which is purely capitalistic. We mostly live in **mixed** economic systems, a mix of capitalistic and socialist rules, so we don't starve to death if we are poor or have to redistribute all our wealth if we are rich.