Learn Russian: Мой брат учит китайский язык.

Yes, they are very similar! Учить is common in everyday speech for learning a specific subject, like a language or a song. Изучать sounds a bit more academic, like studying a major at a university in depth. Here, учит is the perfect everyday choice.

Why does the English translation say "is learning", but Russian uses just one word, учит?

Russian does not have a separate continuous tense like English does. The single present tense verb учит can mean either "learns" (as a general habit) or "is learning" (right now), depending on the context.

Wait, shouldn't the direct object "Chinese language" be in the accusative case here? Why don't китайский язык change their endings?

They are in the accusative case! For masculine, inanimate nouns (non-living things) like язык, the accusative case looks exactly the same as the nominative (dictionary) form. The adjective китайский also copies this rule, so the spelling stays exactly the same.

Could I just say Мой брат учит китайский without the word язык?

Yes, in casual conversation, Russians often drop the word язык (language) and just say учит китайский. However, keeping язык makes it a complete, standard sentence that is great for beginners to practice.