You don’t have to include “book” a second time in the sentence to make it correct.
"Raj's book is better than Julia's" is also a correct sentence for this question "राज की किताब जूलिया की किताब से बेहतर है।" .
but it says it is wrong.
I think because in the sentence the book is mentioned again so we should also mention it again.
Is बेहतर an English loan word? If so, is there no Hindi equivalent?
बेहतर is actually a Persian loanword.
Correct me if I'm wrong, but I think you could also say किताब से अच्छी है at the end, and the meaning would be the same.
Are they cognates, or is it just one heckuva coincidence?
From what I have read, they are not cognates. The English "better" stems from PIE bhād- ('good', 'better') while Persian "behtar" stems from PIE wesu- ('good'). However, I'm hardly any expert on PIE linguistics so don't take my word for it.