The sentence "我会中文" must be translated to English as "I know Chinese", or "I can [a verb such as speak] Chinese", but not "I can Chinese", because "会" is used as a regular transitive verb, a usage not existing for English "can". In the first translation here, "会" matches "know". But if you mull over the connotation, there's a subtle nuance that easily escapes our attention. To know is to have knowledge. "I know Chinese" implies that I have knowledge of this language, a passive knowledge not readily leading to an action. The Chinese "会", on the other hand, often suggests a more active role, and "我会中文" is more accurately translated to "I can [a verb such as speak] Chinese" than to "I know Chinese". The only problem with this "more accurate" translation is that we can't assume "会" is unambiguously "can speak"; of the various aspects of the language skill, speaking is only one, parallel with reading, writing and listening comprehension.
There seems to be a deficiency in second language education in China when compared to that in other countries. "哑巴英语" (literally, "mute or dumb English"), referring to English education with emphasis on scoring high on paper tests at the expense of speaking skills, was and probably still is widespread in China. But language study in other countries is generally in a better shape, where someone said to know a language is assumed to be able to speak that language. As a result, "我会中文" and "I can speak Chinese" become equivalent in real-life situations.
It's obvious that Chinese "会" is used as an auxiliary verb when it's followed by a regular verb, just like English "can". When "会" is followed by a noun, a usage missing for English "can", it is a full-fledged regular verb. In this sense, "会" means "be capable of" or "know" as in "know a language". The noun that follows must represent a type of skill. A language is probably the most common example. But many other skills work as well, e.g., "他会魔术" ("he can do magic", "he knows how to perform magic"), "他会书法" ("he can do calligraphy", "he's good at calligraphy"), "他会量子力学" ("he knows quantum mechanics", although this English sentence may be better interpreted as "他懂量子力学"). In other cases, it becomes ambiguous whether the object is a noun or verb, e.g., "我会游泳" ("I can swim", "I know how to swim"), where "游泳" can be both a noun and a verb.
Chinese is not the only language where the verb "会" may function not only as an auxiliary verb but also as a regular verb. In the Facebook Polyglots group, one German learner asks, "Why do I come across sentences where the main verb is left out; 'Ich kann Deutsch auch'....Where is 'Sprechen'?!". That's simply because the German word "können" (for which "kann" is the first person singular form) serves as a regular verb here. Interestingly, the question asks "Where is 'Sprechen' [speak]?", consistent with the above observation that "speaking" is the dominant or default aspect of the language skill.