I don't want to disagree with your nuanced and well-reasoned reply in general.

However, ownership exists in egalitarian forms of organisation too. It's not the "exclusive right to prevent others from using the thing", but in any human arrangement people understand that some objects are needed by some people (crutch, broken leg), some objects are more actively used by some people (household hatchet, cutting wood every week), and some objects have sentimental value for some people (grandma's ashes.)

These things normally count as property, with no violence to enforce it.