A word that entails the meaning of "the trait of being public"

'publicness' is a word. It sounds alright to me, too.

https://www.dictionary.com/browse/publicness

If something is not owned by anyone in particular, we say it is 'public property' or in 'public ownership'.

It rather depends on context but if the primary idea is of something not being private property, then Commonality

https://www.etymonline.com/word/commonality

commonality (n.) c. 1300, "the people of a country, a community," from Old French comunalte, from comun (see common (adj.) as if from Medieval Latin *communalitas. A respelling of commonalty (late 13c.). Meaning "the common people" is attested from 1580s; that of "state or quality of being shared" is from 1954.