Is a husband to offer a sacrifice on behalf of his family or on behalf of his wife, knowing that they are in two different countries? And is he required to offer two sacrifices, or should his brother offer a sacrifice on behalf of his family, and he offer a sacrifice on behalf of his wife?

The Udhiyah (sacrifice) is a confirmed Sunnah. A single sheep is sufficient for a man and his household, even if their number is large. The household includes the wife, children, and relatives living in the house who are supported by the head of the household or share in his expenses. However, for someone who lives in a separate house or has independent financial support, their participation in the sacrifice is not sufficient. Therefore, it is not enough for your husband to offer one sacrifice for his household and his mother's household. Rather, he should sacrifice for himself and his household. If he desires reward and righteousness, he should sacrifice a separate animal for his mother's household, but if he does not, there is no blame upon him.