Out of 195 US recognized states, it seems there are these types, based solely on names:
- 81 use the word Republic.
- 9 use the word Democratic, but all of those also use the word Republic
- 2 use the word Socialist, but again all also use the word Republic, and 1 uses the word Democratic.
- 18 use Kingdom
- 3 use Principality
- 1 uses Grand Duchy.
- Independent or cooperative
- 3 use Commonwealth, which is ignorantly associated with Socialist at times, but is better understood as "once influenced by England".
- 10 use some form of Federal, including 1 using Confederation, with 6 of them also using the word Republic.
- 7 use the either Union or United, with 1 using Republic and 1 using Kingdom.
- 9 use the word State.
- 2 use the word Independent, both using the word State as well.
- 5 use the word People's, all of them also use Republic, and 3 of them also use the word Democratic.
- 4 use the word Islamic, with all of them using the word Republic again.
- 1 uses the highly religious word Holy
- Most of the rest do not use any ideological words in their names.
If you look closely at any of those in the list above, there is no rhyme or reason why some of them would use the words they use in their official long names. There is no consistency in the use of these words in country names. Especially with the word Republic, some of these names do not accurately describe the the ideology of the state's government at all.
For example, take any of the Independent states under item 3. Many of them are quite capitalist, but the first thought for any of them when you see their names is probably "free from authoritarian government", not "capitalist". While at the same time, many of the Kingdoms are not the typical authoritarian types we'd imagine. Some of them are impeccably free, and some are more capitalist than others by a large degree.
But to actually answer your question, why do none of them use the word Capitalist? As others have said, capitalism is not viewed as much ideology as some of these other words. It's far more associated with economy rather than government, while many of the other words listed above are very much associated with government ideology. The word Capitalist comes from the word capital, as in money and property, and such was the limit of its use until 1850. Much of the modern meaning's early use was made by communist and socialist societal critics. These critics created ideology that centered around Socialism and Communism. Eventually, these ideologies bore new countries, and the names reflected that. Capitalism never bore a new country. It seems more of a default position. No revolution was fought and won over a communist state then reorganized and renamed as a capitalist state. Renaming "Capitalist Republic of ..." would make little sense. If the trend to use the word Capitalism as an ideology (perhaps defined mostly by minimal governmental interference), then maybe new states in the future will take the name, but it seems the Independent category already fills that need.
With all this in mind, it seems that naming countries descriptively is a failed practice. Perhaps naming a country is more of a populist effort. So even if a country did use the word Capitalist in its name, I wouldn't take it any more seriously than the word Republic.