No, this is absolutely wrong. Docker containers don't contain a kernel or any init system. They are simply Linux processes that are isolated from each other using a few native features of Linux.
Running an Ubuntu user land on top of some generic other Linux kernel will for almost all practical purposes feel the same as running real Ubuntu.
That kernel is what is shared between different containers. You can have 5 copies of Ubuntu (or Debian, or Mint) in separate containers, but they're all using the same kernel. This is what makes containers much more efficient than multiple full VMs.