The tech and some of the infrastructure was, but the cultural phenomenon that people are nostalgic for was not, that sprang up organically.

ARPAnet definitely wasn't built "by" the military.

"For" the military is close enough, although "defense research" would probably be more accurate than "military" given that initial users were centered around large universities.

From the start the DoD had issues with the free and open culture from the userbase that skewed towards academic types, later splitting off into a restricted MILNET after a few years.