Repeated elsewhere, as well.
The assertion of the uniqueness of the Office of the Vice President (OVP) is old news, mentioned in 2004 in the 'United States Government Policy and Supporting Positions' (Plum Book):
APPENDIX NO. 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. The Vice Presidency performs functions in both the legislative branch (see article I, section 3 of the Constitution) and in the executive branch (see article II, and amendments XII and XXV, of the Constitution, and section 106 of title 3 of the United States Code).
The annual legislative branch appropriations act (see, for example, Public Law 108–83) and the annual transportation-treasury appropriations act (see, for example, Public Law 108–199) provide funds for the Vice President to hire employees to assist him in carrying out his legislative and executive functions. Executive branch employees also may be assigned or detailed to the Vice President (see 3 U.S.C. 112) and the Vice President may employ consultants (see 3 U.S.C. 106(a)). The Office of the Vice President (OVP) consists of the aggregation of Vice Presidential employees whose salary is disbursed by the Secretary of the Senate from the Vice President’s legislative appropriation, Vice Presidential employees employed with the Vice President’s executive appropriation, employees assigned or detailed to the Vice President, and consultants engaged by the Vice President.
The numbers, titles and salaries of OVP personnel change with some frequency. The salaries of Vice Presidential employees whose salary is disbursed by the Secretary of the Senate from the Vice President’s legislative appropriation cannot exceed a maximum specified by law (see 2 U.S.C.60a–1). The salaries of Vice Presidential mployees whose salary comes from the Vice President’s executive appropriation also cannot exceed a maximum specified by law (see 3 U.S.C. 106).
The authority to appoint, administratively determine the pay of, and discharge Vice Presidential employees rests with the Vice President.The Office of the Vice President is listed under the Executive Branch heading in the online Plum Books for years 1996 and 2000, but not 2004, the year in which the above excerpted statement appears.
More here, from ABC's Justin Rood.