There's some special cases with mountpoints. As far as I remember if you mount something, some things come from the mointpoint, not necessaritly the parent directory?! But I wonder what's wrong here. I'm fairly sure you should be able to create directories inside of a directory where you have write access. No matter if it's for another group etc. I can do it on my computer.
You can also do findmnt /data/Files to see if it's for some reason mounted read-only... And id username should show you if the user is really part of that group.