I think this is the same construction as 'Is leamsa ...' ('... is mine') in Scottish Gaelic. "'S ann leamsa a tha ..." would be equivalent to " ... belongs to me", which is subtly different in English, but I'm not sure there's a distinction between the two constructions in Gaelic.
It's my book is more specifically Is é mo leabhar é, but context is important. If you were saying "It's my book, I've been looking for that everywhere", you'd say Is é mo leabhar é, bhí mé á lorg ar fud na háite. On the other hand the answer to "Whose book is this?" - Cé leis an leabhair seo? is Is liomsa an leabhar, which can reasonably interpreted as "the book is mine" or "it's my book", and you could even interpret the question and answer as "Who owns this book?" and "I own the book" ("own" can be a verb in English, but there is no equivalent verb in Irish).
So you're not wrong, but the phrase "it's my book" is ambiguous in English, and can mean two different things.