There is a weak sense in which the Alcubierre geometry is a "solution" of general relativity. But anything is a "solution" of general relativity in this same weak sense. General relativity says that the matter distribution in a spacetime is related to the geometry of the spacetime in a certain way. You can take any spacetime geometry (as long as it's twice differentiable), plug it into that equation, get a matter distribution, and then say "if only we could make this matter distribution happen, we could make this spacetime geometry happen", with exactly the same plausibility as the Alcubierre geometry.
For example, suppose you want the Sun to suddenly disappear. I don't mean that it accelerates away, or blows up, or anything like that. It just disappears. Its gravitational field goes to zero, planets fly away in a straight line, etc.
It's easy to do this. You just take a spacetime metric with the Sun in it (the Schwarzschild interior and exterior solutions stitched together), and a spacetime metric without the Sun (Minkowski space), interpolate between them with a smooth (or at least twice differentiable) function of time, and plug that into the Einstein equation. The result will not make much sense: you'll find that the Sun's mass flows out to infinity through a region with zero mass density. Exotic matter to the rescue! By introducing negative-mass matter, you can counter the mass of the outflowing matter and make the total mass density zero in the region where general relativity says it has to be zero.
It's important to understand that this matter isn't "exotic" just in having negative mass. It's exotic in that it doesn't follow any physical laws. It just shows up out of nowhere (literally, out of vacuum) during the disappearance of the sun, then disappears into nowhere. It isn't even subject to cause and effect, much less any more specific physical theory.
The same is true of the exotic matter in Alcubierre's spacetime. It is not a "warp drive", because that would imply that the exotic matter could come from the spaceship that's going to ride the bubble. It can't, because the exotic matter on the outside of the bubble is spacelike. That means that either it travels locally outside the light cone, or it arises independently everywhere along the path. The first case would make the solution uninteresting, since if you can travel outside the light cone then you don't need a general-relativistic warp drive to circumvent light speed. So the exotic matter can't come from the ship. In Alcubierre's geometry, it appears miraculously out of vacuum just before the ship arrives, then disappears into vacuum after it leaves. General relativity is perfectly fine with this. As soon as you add any additional physical laws, it's ruled out.
People have speculated about exotic matter guns that could pre-distribute exotic matter with the right properties. You could likewise speculate about a Sun-vanishing gun. The argument in both cases is "Well, based on everything that we think we know about the world, this is impossible. But if you ignore some of that, the remaining premises aren't strong enough to prove it's impossible any more. So maybe it's possible!" Yes, maybe. All science is subject to revision. But based on everything we know right now, the Alcubierre drive is as scientific as time travel. (In fact, it could be used for time travel.)