According to Catholicism, is the type of love shared between all heavenly members an affectionate brotherly love mentioned in the Bible?
The short answer seems to be yes, at least for those we have a certain affection for.
St. Thomas Aquinas answers this question as such: Whether the order of charity endures in heaven?
The order of charity must needs remain in heaven, as regards the love of God above all things. For this will be realized simply when man shall enjoy God perfectly. But, as regards the order between man himself and other men, a distinction would seem to be necessary, because, as we stated above (Articles 7 and 9), the degrees of love may be distinguished either in respect of the good which a man desires for another, or according to the intensity of love itself. On the first way a man will love better men more than himself, and those who are less good, less than himself: because, by reason of the perfect conformity of the human to the Divine will, each of the blessed will desire everyone to have what is due to him according to Divine justice. Nor will that be a time for advancing by means of merit to a yet greater reward, as happens now while it is possible for a man to desire both the virtue and the reward of a better man, whereas then the will of each one will rest within the limits determined by God. But in the second way a man will love himself more than even his better neighbors, because the intensity of the act of love arises on the part of the person who loves, as stated above (Articles 7 and 9). Moreover it is for this that the gift of charity is bestowed by God on each one, namely, that he may first of all direct his mind to God, and this pertains to a man's love for himself, and that, in the second place, he may wish other things to be directed to God, and even work for that end according to his capacity.
As to the order to be observed among our neighbors, a man will simply love those who are better, according to the love of charity. Because the entire life of the blessed consists in directing their minds to God, wherefore the entire ordering of their love will be ruled with respect to God, so that each one will love more and reckon to be nearer to himself those who are nearer to God. For then one man will no longer succor another, as he needs to in the present life, wherein each man has to succor those who are closely connected with him rather than those who are not, no matter what be the nature of their distress: hence it is that in this life, a man, by the inclination of charity, loves more those who are more closely united to him, for he is under a greater obligation to bestow on them the effect of charity. It will however be possible in heaven for a man to love in several ways one who is connected with him, since the causes of virtuous love will not be banished from the mind of the blessed. Yet all these reasons are incomparably surpassed by that which is taken from nighness to God.