Just as a native speaker, not as an expert on these sorts of structures, when I say or hear a possessive like Tom's, I process it the same way I would process "my/mine" or "your/yours". Your book: a book that belongs to you, or a book of yours Tom's book: a book that belongs to Tom, or a book of Tom's But I never feel like there's a silent piece I'm omitting, so for example if I were to say "a book of Tom's", I don't feel like I'm starting with "a book of Tom's books" and dropping "books" from the end. It's more as though I think "of Tom's" as more simply meaning "that which belongs to Tom" or "that which is associated with Tom", etc.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.