Earlier quoted context omitted.
I'm sure they did, as they both would have taken the exact same course of action. The PATRIOT Act itself was written by Sen Joe Biden ffs.
The same course of action? War with Afghanistan was certainly inevitable after 9/11 (assuming Gore, like Bush didn't follow up on the warnings, and the attacks still happened). The Iraq War, on the other hand, would not have happened.
Regime change in Iraq was the official policy of the Clinton administration, President Clinton initiated frequent air strikes and other military action against Saddam, Vice President Gore criticized George HW Bush for not removing Saddam in 1991 when he had the chance, and Sen Hillary Clinton voted to authorize the war in 2003 (while she is certainly an independent person, they have been very closely aligned in policy, both Hillary and Gore were/are self-styled continuations of Bill Clinton's Presidency), and there was broad support for removing Saddam from power even before 9/11.