A true antonym is a word opposite in meaning.
A brother and a sister are both relatives, or siblings, so I agree that in the context of discussing relatives they are not really opposite in meaning any more than cousin is the opposite of uncle - they are simply different relatives.
However, "brother and sister", like "male and female" are traditionally binary choices or binary options, meaning that if something is not one, then by default it must be the other. If you have a sibling it is either a brother or a sister (this is of course barring any modern view on the number of genders and I ask that if you feel strongly about that matter you do not allow it to cloud your view of how a question about English should be answered).
So, given a choice of two things, where it can only be one or the other, the two are technically opposites and therefore antonyms.
For example, "up" is obviously the opposite of "down". However, there are other directions such as left and right. They all have contrary meanings, but directionally there is an opposite relationship between left and right, and between up and down, reducing the 4 directions to two binary choices and two sets of anyonyms.
I would have to agree with the dictionary definition that "brother" and "sister" are antonyms when considering the (traditionally recognised) gender of siblings.