The bill is not listed as a companion bill to the one introduced by Paul and Sens. Ted Cruz (R-Texas) and Mike Lee (R-Utah), although it appears to have the same goal as the Senate bill. That bill, S. 505, explicitly prohibits the use of drones to kill U.S. citizens.
Cruz, the lead sponsor of the Senate bill, introduced it a day after Paul's March 6 filibuster on the Senate floor, in which he called on the Obama administration to clarify its policy regarding domestic drone strikes. On March 7, Attorney General Eric Holder wrote to Paul to say the government does not have any authority to use drones against U.S. citizens on U.S. soil that don't pose an immediate threat.
Paul said he pressed the issue after Holder seemed to waffle in answering a question about whether these sorts of drone strikes are legal. But Paul's filibuster was criticized that week by Sens. John McCain (R-Ariz.), who accused Paul of doing a "disservice" to Americans by "making them think that somehow they're in danger from their government."