Translation:I recognize your mother and your brother.
“Know” has many meanings, and its “recognize” meaning would be an accurate translation for aithin. Other senses of “know” might not use aithin, e.g. “X knows Y” in its “X is acquainted with Y” sense would be expressed as Tá aithne ag X ar Y. Given the lack of context, “know” should be accepted, but it should be realized that it’s only correct for certain meanings of “know”.
No - bráthair isn't part of Duolingo's vocabulary, and is not used to describe siblings, so it would be very out of place in an exercise that also referred your mother. (The bhr sound in bhráthair would also sound quite different).