"The brother is studying" would work. It accepted it for me. It is also correct English.
But "The brother does study" would only be correct if you are doing something like a comparison as in "The sister is not studying but the brother does study. It is more of an emphasis that he is indeed studying.
For family relationships, you can drop the articles to imply that it is your relative.
For example, I can say "Father studies" or "Mother studies". It sounds a bit old fashioned and formal, but the implication is that it is the speaker's mother/father.
Similarly, I can say "Brother studies" or "Sister studies" with the implication that it is my brother or sister. Again, it sounds formal and old fashioned, but it is grammatical.
This construction is used extensively in the Berenstain Bears books: https://www.putmeinthestory.com/media/catalog/product/cache/1/image/9df78eab33525d08d6e5fb8d27136e95/t/h/the-berenstain-bears-pet-show-6.1561153748.jpg
As you said, the implication is the speaker's mother or father, but here, it's "the brother", we don't know whose brother it is.
I can't find nowhere in English grammar books this "zero article" used to replace a possessive. I think it's not official grammar, and the example you gave is maybe related to something a little babyish (?).
But if you find something about that, I'd be curious and interested by the reading.
Now: "Brother studies" is accepted, but I'm not very sure it's good English grammar (but I'm not a native)
Edit: I've found that this zero-article was used for "titles", like "mother" considered as a title for instance.
I'm confused as well. Are definite articles mandatory regardless of context? Because as I understand it, in this context it could go either way in English. It is stilted to say "brother studies," but not only is it still grammatically correct, the course has accepted things like "Sister is in the city and Mother is at home."
Granted, it is in Beta still, so either way this appears to be a bug of some kind.
No, it's not proper English grammar.
Have a look here about the use of the "Zero article" in English:
Other than memorizing, and repeating it, if your first language is English, it may help to remember that words like fraternity, fraternize... and such are related to "frater"/brother, and paternity to "pater"/father. "Papa" a form of "dad/father" also sounds sort of close to "pater". Or at least it starts with "pa"
Personally, I tend to just try to learn by memorizing and repetition since relying on similarity can end up being really misleading. But quite a few people seem to like doing it. (So it may help a bit if you are careful.)
I gave them the answer they want, but it is something I would never say. " The brother studies." No. Never. "Brother studies." would be a stretch, but more natural. In a regional way, a person would say, "Bubby is studying." or "Bro is studying." Basically, Brother is being uses as a name.