The informal, conversation verb 'swap' can have more than one meaning.
Swap almond milk for regular milk.
Here, "swap" means "substitute". We substitute the second thing for first thing.
He swapped his cupcake for a candy bar.
Here, 'swap' means 'exchange'. We exchange the first thing for the second thing.
My question: Is there a correct interpretation if there is no possessive attribute?
The correct interpretation is obtained by thinking about what is being described.
He swapped a cupcake for a candy bar. What does this sentence actually mean?
Previously you used 'his cupcake'. The word 'his' made it clear that the person initially possessed the cupcake, thus the 'swap' is an exchange. Now you use 'a cupcake' and it is no longer clear. The phrase is ambiguous. It may or may not be possible to resolve this by examining the context (the surrounding words).
It could mean either "He had a cupcake and gave it up in exchange for a candy bar." or