No, product managers do not get to claim they made something.
They led a team that made something.
They led a team that made something.
Do I get to claim I made the Python TheseusA, or TheseusB, or neither or both?
I think product managers surely have a right to claim they "helped" to make something at least. If the team made something then those who work on the team all assisted in "making" the thing.