The story
From the study of generators. The motivation for calling topologies satisfying sense 1.5 "finitely generated" is that any topology satisfies sense 1.5 if and only if it is coherent with its finite subspaces. Thus, metaphorically, it is "generated" by them. The category-theoretic senses were created to generalize those of abstract algebra, and so were named identically.
Etymology from Wiktionary, CC BY-SA 4.0.