Why sister [nouns] and not brother [nouns]?

In describing relations between entities as if they were relations between people, you are personifying those entities. "Sister", "mother", and "daughter" are common because personifications are usually female.

The predominance of females is at least partly because Latin grammar gives nouns for abstractions the female gender.

The Oxford English Dictionary shows that this usage goes back to at least the 1500s. It provides the following definition:

Appositive, with the sense ‘fellow’, ‘having a close kinship or relationship to another’; ‘belonging to the same class or group’. In linguistic contexts, denoting a parallel familial relationship (as between languages, dialects, etc.). With reference to places, institutions, etc., sometimes implying a more or less formal link.

Here are some examples of early usage:

• 1570 J. Foxe Actes & Monumentes (rev. ed.) I. f. 363v/1 A sister Church one with an other, seekyng together ye glory of Christ.
• 1611 J. Speed Hist. Great Brit. ix. x. 546/1 The right of our noble sister nation.
• 1641 J. Milton Of Reformation 70 We must..come from Schisme to unity with our neighbour Reformed sister Churches.
• 1679 J. Fell in J. Gutch Collectanea Curiosa (1781) I. 270 If we are justified, the advantage will extend to our Sister University.

This doesn't answer your question as to why "sister" is used, although if the usage was influenced by French or Latin, "church", "nation" and "university" are all feminine nouns in those languages.