The short answer is "no."
The long answer is a (to most people) really boring history lesson about Senate rules, and the creation of "cloture."
So here's the medium length answer. A long time ago, a change to the Senate rules created a loophole, which allowed Senators to drone on and on endlessly and run out the clock on legislation. They could read from the phone book, or from the bible, or rant in paranoid fashion about how some day there would be a black president, and he would try to raise taxes on rich people. Whatever.
It looked a lot like what Bernie Sanders is doing today.
But when Senators began to abuse the rule, the Senate created another rule to allow a supermajority of members to vote to cut off debate. That's what happens in the current era when the Senate invokes "cloture," and what will almost certainly happen with the tax cut debate.
The consequence of that change was pernicious, though. It meant that a minority of senators could very passively block legislation, simply by saying they'd vote against cloture. That's what happened with health care reform. The "filibuster" of health care reform was, in essence, the several weeks between when Harry Reid introduced the bill, and when he finally secured his 60th vote.
But in the meantime, you didn't see Republicans boldly and tirelessly hold the floor to prevent passage. They just made it clear that they'd never vote "yes".
So even though what Sanders is doing now looks like a filibuster, and is in the spirit of the filibuster in its early days, and is what most laypeople think of when they hear the word "filibuster," it is not a filibuster. And it won't be unless he can keep Reid from rounding up 60 votes next week -- in which case he wouldn't need to be on the floor in the first place.