I have solved this issue! (After a lot of research). See the (rather lengthy) answer below.
Note: I believe this solution will also work for ntfs file systems, which have the same 'quirk'.
Problem:
This issue has all to do with user id's and group id's.
An ext4 file system is owned by one user, but its ownership is recognised by that user's user id or uid rather than the user name. It is usually also owned by that user's group, which has the same id (group id or gid, i.e. if uid=1002 gid=1002) In other words, if user 'A' owns the file system on one computer and has the uid 1001, user 'E' on another computer who also has the uid of 1001 automatically owns the file system.
Ubuntu user id's start at 1000, meaning that as I am the only user on my computer, my uid is 1000. On the other computer, however, my uid is 1001. This is why it mounts under the main user on that computer. It is mounting under the same uid.
Solution:
As said above, a file system owned by a user is usually owned by their group. User groups can contain more than one user, so you can add yourself to the group which owns the file system.
You can use:
sudo adduser [user] [group]
or sudo usermod -a -G [group name] [user name]
(For more info on these commands, see this question.)
to add a user to an existing user's group, giving them access to files owned by that user.
Add your user to the group, make sure that the owner's group has full access to the file system, and problem solved!
exfat-utils
into your Ubuntu installations so they will see the drive, but it does allow for files larger than 4GB. – Terrance Sep 11 '17 at 17:00