I think that actually it's quite relevant to consider that social institutions actually are owned (in the practical sense, not in the legal one) by someone, and that it matters by whom and how they are owned.

Yes, there's the common scenario of social institutions that are owned (controlled, funded, etc) by the whole community they're serving. But there are cases also of important social institutions (for example, education) "owned" by particular subgroups of that community e.g. the various historical examples of key social functions being performed by churches, or funded by particular wealthy individuals for their community, or social institutions implemented by a foreign government/elite for the locals (both in the case of conquests and colonial empires) - in which case the ownership mattered quite a lot; because in this case the social institutions run for the local people weren't controlled by the locals and were significant instruments of political and social influence and control by the "owners" of these social institutions. Which has interesting parallels with the current megacorps controlling much of the information flow.