Rocheworlds
A Rocheworld is a pair of equally sized planets that orbit so closely around there common center of mass that their atmospheres touch. At that point of contact the gravitational pull from the two planets cancel. This would give you a microgravity environment. This is the L1 point of the two worlds and thus would have the gravitational gradients you would expect at L1. This means nothing would be stable there, denser then air organisms would fall toward the worlds, and lighter than air organisms would be pushed away from the center of the envelope. But these are minuscule forces that an organism with any level of activity should be able to overcome. It does mean floating islands and masses of inert vegetation are probably out.
On whether this configuration of worlds is possible, this question addresses sharing an atmosphere. To summarize, if the pull of the worlds on each other is to great that they tear themselves apart. There is a very slim margin in which two terrestrial worlds are close enough to share atmosphere but not destroy each other. But we don't care about the surface of the worlds so I think the better option would be two gas giants. Of note is that at the far end in this direction, stars that share an atmosphere, there are observed cases of their existence.
Speculative Environment
As for the biology, this is how I envision the ecosystem working. 801
The amount of sunlight reaching the area of micro gravity is small due to the worlds eclipsing it on either end. You could have the orbital plane of the planets perpendicular to the plane of the ecliptic which would allow more light but I feel would reduce stability in an already tenuous system. Putting the worlds closer to their parent star would also hurt stability. My solution is to cover the gas giants with photosynthetic algae. The high gravity on the worlds themselves will preclude the development of complex life which leaves the algae without any predators so they can flourish. Air currents pass these algae across the bridge between worlds, possibly even concentrating the nutrients. This gives a food and oxygen source for the complex life in the microgravity environment in between the worlds.
As mentioned earlier, no organism here could be inert, every living thing that can't survive the high gravity of the worlds below, or the vacuum to the sides, would need to constantly be making micro adjustments to keep itself near L1. This could lead to a king of the hill type power struggle, where organisms are vying to be closest to L1.
Another possible plot point is the storms in such a system. I don't know exactly what they would be like, but I am sure they would be interesting.
Note: Rocheworlds have appeared in fiction, most notably <PERSON> series which coined the term. Though I am unaware of any cases where the inhabitable micro gravity environment has been a key point. In part because in fiction the pair of worlds are usually terrestrial making the bridge between them much smaller in volume. 801