Mandarin and Cantonese &c. are considerably more similar to each other than Hindi and Telegu &c, so yeah, “Chinese” is a language but “Indian” isn’t. For example, something written in “Chinese” by a native Wu speaker is still easily readable by a native Min Zhong speaker; the characters and syntax are pretty much the same (yeah, traditional v.s. simplified written Chinese are different, but that is not regional, it is more a matter of era/politics). On the other hand, Hindi/Urdu and Tamil use different alphabets with different rules of syntax; a native speaker of one would not necessarily be able to parse the written word of the other.

Total side note, easier to read version of this image here: