Yes, I think the 'duty' that is being referred to is 'dharma' in the sense of 'true identity', and Arjuna's identity is a Kshatriya (warrior), as are all sons of kingly descent; and he's also trained as an archer. He was born and trained to fight.
"Dharma" also means 'truth'; the implication is that there is a kind of falsity in not being true to the identity you inherit, or that has been given to you (e.g. by military training).
FWIW, I don't agree that people get any permanent identity at birth, nor that they can acquire some kind of permanent identity by being trained as a soldier (or a thief, or an artist, or a comedian). Krishna/Vishnu is misleading Arjuna: we're all fluid, we can all change. Krishna knows this, and in fact he shows it, because he changes himself.
[I confess I'm not closely familiar with the BV - I think I read it in my teens. It's not that long, by comparison with other classic Vedic texts; or with western religious tracts, for that matter]