How to mv a folder in Linux retaining its mtime?

POSIX mv doesn't provide any option to ask for atime/mtime preservation, but as the operation is local to a same volume, you can ask cp to use hard-links instead of copying data of the regular files using the -l option:

cp -p -r -l source/date target/
rm -rf source/data

Since only directories and file references will be actually copied, it should go much faster:

For more informations on hard-links, you can consult the corresponding Wikipedia page

As for why subdirectories mtime is being reset with your current solution, it's because you only get and restore the parent directory mtime : touch is not a recursive command.

Another solution may be:

rsync -a --remove-source-files source/data target/