This is mostly to do with the date of Harris' resignation as Senator. She submitted her letter of resignation to California Governor Gavin Newsom on Jan 18th, two days before being sworn in as VP. Recent Vice Presidents who resigned from their Senate seats did so earlier - with the consequence that their appointed replacement assumed their seat before inauguration day.
The most recent VP who swore in their replacement was Charles Curtis, who resigned as Kansas Senator on March 4th, 1929. His replacement, Henry J. Allen, was appointed by the Governor on April 1st, and took the oath of office on April 15th, administered by now Vice President Curtis.
ADMINISTRATION OF OATH TO SENATORS
The VICE PRESIDENT. The Chief Clerk will call the names of Senators elect who did not take the oath of office at the special session of the Senate in March, and they, with the Senator designate from Kansas, will present themselves at the desk and receive the oath of office.
The Chief Clerk called the names of Mr. CUTTING, Mr. LA FOLLETTE, and Mr. SHIPSTEAD.
Mr. SCHALL. I desire that the announcement that my colleague the Senator from Minnesota [Mr. SHIPSTEAD] is detained by reason of serious illness may stand for the day.
Mr. CUTTING escorted by Mr. WATERMAN, Mr. LA FOLLETTE escorted by Mr. NORRIS, and Mr. ALLEN escorted by Mr. CAPPER advanced to the Vice President's desk; and the oath prescribed by law having been administered to them, they took their seats in the Senate.
71 Cong. Rec. (Bound) - Volume 71, Part 1
Below are all the other subsequent Senators who resigned their seats to become Vice President - in each case, their replacement was sworn in before they became VP. Then VP-elect Biden did attend the swearing-in ceremony of Kaufman, but obviously was not eligible to preside.
|Outgoing Senator / VP
||Successor in Senate
||Sworn in on
||Jan 16th, 2009
||Jan 3rd, 1993
||Jan 3rd, 1989
||Wendell R. Anderson
||Dec 30th 1976
||Dec 30th 1964
|Lyndon B. Johnson
||William A. Blakley
||Jan 3rd 1961
||Jan 2nd 1953
|Alben W. Barkley
||Jan 20th 1949
but before the inauguration of Barkley
(see Congressional Record)
||Jan 18th 1945
|20th Amendment changes inauguration
date from March 4th to Jan 20th
||Henry Justin Allen
||April 15th 1929
In conclusion, this is fairly unusual, but only because Senators who are successfully elected to the office of Vice President tend to step down as Senator early enough for their replacement to be sworn in by someone else.