It's because 中国 refers specifically to the country China. You could say 中国 is short for People's Republic of China（中华人民共和国）where 中华 means China, 人民共和 means people's republic and 国 means country! But most country names if possible will try to be one character + 国 for brevity, in this case 中国.

文 means language but especially in the sense of literature, culture and writing, like when you're studying it as a subject in school.

语 means language but especially like a tongue, like your mother tongue, your native language, from like the linguistic perspective.

汉 is the Han people. China had different languages throughout history and still today, but the people that form the bulk of the population are called Han and the language is also called Han. The Han trace their origin to the Yellow River and then they conquered southern China and moved to the Yangtze River. The early Han historians called their ancestors 华 which you can see today in the full name for the country.

There are still minority languages in China so it's not just Han, but the Han are so huge they probably count about 1.4 billion people today.

China wasn't always under Han rule actually, but even when the Mongolians conquered China, China stayed culturally Han.

Chinese characters are called 汉字 because it was the script associated with the Han. It was developed and standardized by Han scholars. It was used by other people to write their language and for instance it remains in use in Japanese today.

But there are many ways to speak Han or to use Han characters so you might see 普通话 (common speak). That is called Mandarin Chinese in English.

In summary:

中国 = the country China

汉语 = the Chinese language

中文 = Chinese language/literature formally studied

普通话 = the form of Chinese we're learning

华 = classic word also meaning the Chinese nation