Yes, it is a subject complement. If you are studying linguistics at university, you may find thoughtco dot com a useful resource. Here is an excerpt from the article "What Are Subject Complements in English Grammar?" [excerpt] The Difference Between a Subject Complement and an Object The Subject Complement is the obligatory constituent which follows a copular verb and which cannot be made the subject in a passive clause: Who's there? It's me / It's I.* She became a tennis champion at a very early age. Feel free to ask questions! The Subject Complement does not represent a new participant, as an Object does, but completes the predicate by adding information about the subject referent. For this reason, the Subject Complement differs from the Object in that it can be realized not only by a nominal group but also by an adjectival group (Adj.G), as illustrated in the previous examples. [end excerpt] There are several related articles on the site. We can create some sentences similar to your example. [The adjective "poor" is a subject complement.] He was poor. He died poor. [He had been a rich man, but he died poor.] He was born poor. [He was born poor, but became a rich man.] [The noun phrase "a dog" is a subject complement.] Rex was born a dog, but he acted like a human. Rex was born a dog, but he became my best friend. [original sentence - The noun phrase "a puppy" is a subject complement.] I was born a puppy. << Semantically odd, unless this text is from a fictional story where the narrator is an animal (such as the famous novel Black Beauty).

A puppy in this sentence would be the object. The subject is "I". You can even substitute a puppy for an object pronoun. I was born it.

no, it's a direct object