Is it possible to launch different Docker containers with different storage drivers? For example, container 1 with AUFS and container 2 with Overlay2?
If you look at the documentation, you'll see that the choice of storage driver is configured at the level of the Docker daemon. All of the containers and images managed by the same Docker daemon will share the storage driver.
This makes logical sense, once you understand how the storage driver works. Briefly, the concept is that the storage driver manages the interactions between the layers of a Docker image. When you pull down an image from a Docker repository, the storage driver configuration on your system determines how it will be stored and managed locally on your system. Different storage drivers handle the implementation differently, and the layers managed by one storage driver will be incompatible with another storage driver. The documentation also notes that changing the storage driver configuration on a system will render the existing containers and images inaccessible.
The choice of the storage driver is dependent upon several factors, such as the level of support offered by your system, as well as the characteristics of your workload.
Finally, if you want to use different storage drivers for different containers, you can look at running multiple docker daemons, either on the same host or on different hosts. The image data and container data cannot be shared between such daemons, because the implementations are incompatible as noted above.