These should be just empty directories.
Should be, but aren't always. I've seen whole partitions mounted there. I've either (as root) unmounted
them or deleted them with no ill effects. But never before has my home name or root showed up, and I'm
hesitant to delete either one.
You can mount stuff there (external harddisks, usb pen drives),
but as long as the directories are empty, you haven't.
Of course I can. But I always explicitly, manually or in fstab, mount to mount points in my home directory.
I don't want some unknown activity mounting my stuff on /media, unless I know why.
It means you have no read permission, because both directories are owned by root and
can only be read by root and its group, not by others (so 750, not 755).
The user access to the directory with the users name is managed by other means
but I'm not sure which ones exactly. It's a special directory because the system mounts there with user perms.
I don't know what "perms" are, but I do not want them there. It's your "other means" that I am trying to discover.
They just showed up a couple of days ago.
What activity triggered their presence? Why are they there now when they never were before?