As VP, Biden did get a tie breaking vote in the Senate, and was considered the President of the Senate. It is an odd role that is not a member of Congress, but in some ways is a member of Congress.
Anyway, anyone can shepherd a bill through Congress. Many times a President will assign the VP, who is already working in Congress, to shepherd important bills, but not always. In this case, Obama had Biden, a VP with a huge amount of experience with Congress, shepherd Obamacare through Congress.
It was a miracle it got passed. Whether you agree or disagree with the bill, you have to admit that it was a huge success for Biden's competency that he got it through.