Nothing is wrong with the mounting options and fstab file, as far as my knowledge, you just have to use below command to get the Required right permissions on the media folder. and I will explain you why.
1) your current permissions on the folder as you mentioned are as below. (Apologize if you know this and I'm bothering you with it.)
drwxr-xr-x 14 root root 4096 Jan 13 19:37 media in this d stands for directory, first group of rwx stands for the rights of the user who owns that object, which is in your case root.
second group of rwx stands for the rights of the group who owns that object & here you have only r-x which means even if you are in the sudoers file or in the admin group you do not have rights to write in it.
Third group of rwx is or other users, who are not owner of that object nor in the group who has ownership rights of that object. if your case it is again r-x only. so that is why you do not have write permissions on it.
2) So you need to change permissions of the media folder. (I am going to consider that you are in admin group or sudoers file, because this will work only in that case). so this is what you need to do.
$ sudo chmod -R 775 /media
3) This will give you required write permissions on the all auto mounted drives.
Please check and let me know if this does not work will tell you what else can be done to fix this issue.