My dad handed me down the same gun.
I think it is the same thing as "My dad handed down the same gun to me" and the original sentence sounds natural.
Is my thinking correct?
Yes.
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Note there are two possible meanings of hand down, as follows.
(A) hand down can mean 'pass ownership from a member of one generation to a member of another generation, eg from father to son.
(B) hand down can also mean to physically hand something to a person at a lower physical level, eg my father stood on a chair and passed something down to me