I feel the opposite. Writing in a book is the final step in really making it mine. Not writing in it seems to be an insult to the author, that what they wrote isn't important enough to engage with (though some are that bad).
Thinking about my childhood, the vast majority of the books I read were from libraries. School books were also all lend from school. Sure, I also owned plenty of books myself but my parents would have been pretty pissed if I vandalized them as my siblings might want to read them.
For me books where always something to share not something to own. Never got the need to make them "my own".
Honestly, when people who take care of their things die, it's like an explosion of wealth. The things in my place range from the mid-19th century to today. If you let people pick through my shit and take something that looks cool, my death could result in 1000 smiles.
edit: I even take care of the things that I've made. They're no doubt mine, but some of those things will make someone else just as happy as they've made me.