It's possible, maybe, with some caveats. If the planets get too close to each other, there is a high likelihood that, over time, one of them would "clear the neighborhood" of the other: it would capture it or fling it off into a distant orbit. This is a big reason why planets are distantly spaced from each other: the planets do perturb one another in their orbits, but if the gravitational pull isn't enough to capture the planet (over time), they tend to get in a particular sort of synchrony or resonance that keeps them mostly stable. So that's a prerequisite for this to happen: the planets must be stable over time.
The second thing that would have to happen is that the moon would have to orbit with a high eccentricity such that it gets far enough away from Planet A to be captured by Planet B, and vice versa. An orbit with a low eccentricity is likely to either get caught in the planet-clearing scenario above or not have the moon wander far enough away from Planet A to be captured.
Finally, the orbit must synchronize well enough to keep the orbits stable: the energy exchanged by capturing the moon must be given back completely on each pass and at the exact right time, otherwise, the orbit will decay and circularize enough that it eventually becomes captured to a single planet. This is the point at which this becomes sketchy: I expect that this sort of perfect synchrony is unstable over long time periods due to the chances that any little gravitational perturbation will cause the moon to fall out of being in the perfect place at the perfect time. Not only does the moon have to encounter stable planetary transfer, but the orbits of the planets themselves must be stable during the time they have the moon and the time when they do not (remember: planets and moons orbit each other around their epicenter, so the loss of the moon would alter the planet's orbit). That's an awful lot of orbital perfection that, if one tiny thing does not line up, comes crashing to the ground, maybe literally depending on how the system becomes unstable.
So, without running simulations, I would suspect that it's possible, but really highly unlikely to happen naturally, becoming unstable over long time periods. However, we could do something like this with an artificial satellite which can correct for instabilities.
As an interesting real-world application of this concept, consider the Mars Cycler that Buzz Aldrin proposed. There are a number of possible orbits that work for this, most of them do not propose having an orbit within one or the other planet (it's a slingshot-to-slingshot sort of idea), but some of them do. Warning: that page will automatically download a small MP4 video of a proposed spacecraft, but that's fairly innocuous. Look at the diagram in white part-way down the page: this is a more exotic cycler orbit that involves a loopy orbit around Mars that has a gravitational capture, a pass around the planet, then a gravitational slingshot back to Earth.
This is sort of what the moon-slingshot idea would look like.
