Urban dictionary was no help on the 3 book words, except for this bit of localized British slang:
Merriam-Webster’s entry for booky just says it’s a synonym for bookish.
The “Book ‘Em Danno” meme doesn’t seem helpful in deciphering Lawton’s wordplay, either.
Comics I Don't Understand 
I think it’s just a novel bit of anthropomorphic surrealism (a “coming-of-age” story).
P.S. The problem with looking for an explanation in British slang is that Mary Lawton is from Houston.
Something that’s X-y is slightly more X than something that’s X-ish. Start with the perfectly normal word “bookish” and you get this sequence.
Yeah, I think it’s a sequence like DemetriosX says but I can’t think of any adjective that works with both -ish and -y as mitigators.
Bookish: A young reader with glasses that would rather read then go outside.
Booky: A fan of Books, no age or gender?
Fully Book: Mature reading matter?
When it is short for bookmaker (the fellow who takes bets), I think bookie is spelled only that way, and would look odd as booky.
Back in the early early days of Amazon, there was a reviewer I recall who would troll the review pages of various books with 5 star reviews exuberantly describing them as “Verrrrrrrry Booky!”
I laughed back then, but I was a much younger fellow.
@ billytheskink (6) – I went looking for an example, and discovered that those stupid trolls are still around: