File path with space as variable in bash script

It's not a problem with variable assignment, you're doing it correctly. It's how you're using it.

This:

my_path = "/some/path with/space"
docker run --rm --mount source=$my_path,target=$some_other_path,type=bind …

Will, after substitution, become this:

docker run --rm --mount source=/some/path with/space,target=/some/other/path,type=bind …

You need to quote the argument to the docker command during invocation:

docker run --rm --mount "source=$my_path,target=$some_other_path,type=bind" …

Alternatively quote just the variable. Bash merges adjacent strings, quoted or not.

docker run --rm --mount source="$my_path",target=$some_other_path,type=bind …