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: 
Short answer : because capitalism is a **transversal** marker and because most countries **take for granted** that they're capitalists. Why would you need to state that you're capitalistic when you feel like it's the default setting?
(so much so that they sometimes don't even realize that they are -- just like you when you write "the US are not so much of a capitalistic country", which is hilarious).

So they focus on **other aspects** of their ideology: for example, within the world that relies on markets, they feel like their differences come from something else, such as their type of government (republic versus monarchy versus whatever) or within the muslim world they focus on their religious families (are we more sunnites or more hiites or etc.)

I would add that **most countries switched to capitalism without realizing it**. Capitalism as we know it is the result of moving away from manufactures (think of them as half-baked factories relying on large-scale craftsmanship and royal approval+funding). In countries such as France in happened **alongside another big change** which was more visible and more easily labelled (such as the Revolution and the institution of democracy as a Republic). It took over a century for people like Marx to pinpoint what actually happened, from a **societal** pespective; People like Adam Smith had analyzed very early the economical aspect of things but didn't see how much capitalism is intertweened in the philosophy and government system.
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.