Take a look in the /etc/passwd and /etc/group files and see what the userid numbers are assigned. In the passwd and group file, the root user is normally 0. In the group file, the users group is normally 100. (Note the "s" on users) I have not seen a user in the passwd file called "user" before. But if the login that you are actually using has an ID or group ID set for another account then that account will be used.
The very first user added to the passwd file will usually start with the ID of 1000 and in the group 100. The group 100 would be the users group. But if the login userid was modified to 0, then that user would be in group assignment for root. Perhaps you normal login account is using the name "user" and that ID might be correct, but it might just be in the wrong group.
If you really want the user ID's and the groups ID's they are set but you want that user to be able to modify the files by the group permissions then you might try changing the umask value assignment in in /etc/profile to 002 instead of 022. That would allow owner and group full permission for any new files created and world read and execute only. For the umask, value it is exactly opposite what you desire for the settings. I think the operation is called a bit level "NOR" operation. The full permission mask of 777 is NOR'ed with the default umask to clear the bits that are set in both values. Anybody else can feel free to correct or elaborate more if desired.