With every new administration, the government publishes what known as the Plum book. The Plum Book is used to identify presidentially appointed positions within the Federal Government. I had a look at the latest one and notice there was a separate appendix entry for the Office of the Vice President. This entry was also in the 2004 Plum book. Prior to that, the Office of the Vice President was part of the Executive Branch just like the Constitution says.
Appendix 5 - Office of the Vice President
The Vice Presidency is a unique office that is neither a part of the executive branch nor a part of the legislative branch, but is attached by the Constitution to the latter.
So what did your bullshit detector do?