Here is the truth you are looking for.
The Secure Fence Act of 2006
The Secure Fence Act of 2006, which was passed by a Republican Congress and signed by President George W. Bush, authorized about 700 miles of fencing along certain stretches of land between the border of the United States and Mexico.
The act also authorized the use of more vehicle barriers, checkpoints and lighting to curb illegal immigration, and the use of advanced technology such as satellites and unmanned aerial vehicles.
At the time the act was being considered, Barack Obama, Hillary Clinton and Chuck Schumer were all members of the Senate. (Schumer of New York is now the Senate minority leader.)
Obama, Clinton, Schumer and 23 other Democratic senators voted in favor of the act when it passed in the Senate by a vote of 80 to 19.
Originally, the act called on the Department of Homeland Security to install at least two layers of reinforced fencing along some stretches of the border. That was amended later, however, through the Consolidated Appropriations Act of 2008, which got rid of the double-layer requirement.
Currently, 702 miles of fencing separates the United States from Mexico, according to U.S. Customs and Border Protection.
The Truth is only 23 Democrats approved of the fence part of the 2006 Border Patrol Act- Nancy Pelosi was not one of the Democrats that agreed to the "DOUBLE" Fence part of the bill, so she voted against it.
In 2008, another Bill replaced the 2006 bill with a Bill that deleted the "Doubling" of the fence- Nancy and other Democrats did sign onto the 2008 Bill.
SO, Mulvaney is wrong to say that the idea of a WALL had full unanimous Democrat support in 2006.