After successfully installing Docker in WSL, if I run standard Docker commands, I get connection errors. Runing with sudo or not yields the same result in all examples.
As other answers stated, Docker can't be installed and hosted inside WSL but can be used directly from there as any Windows executable.
It may only require an extra effort to make it usable without those pesky .exe extensions:
Bash on Windows - alias for exe files.
The linked workaround provides a ready to use custom script for easy symlink creation. Symlinks can be freely used inside non-interactive shell scripts unlike bash aliases.
There is also an approach with connecting from WSL to Docker for Windows as remote host described in https://nickjanetakis.com/blog/setting-up-docker-for-windows-and-wsl-to-work-flawlessly.
If you’re wondering “why not just run docker.exe and docker-compose.exe from Docker for Windows directly in WSL?”, that’s due to a bug with running Docker or Docker Compose interactively in that environment. The TL;DR is you can’t run anything in the foreground with interactive mode, which makes it unusable for real web development.
But sadly there is no reference to exact bug symptoms there. And I didn't face it so far.