why does my Time Machine backup require more than 2x the gigabytes my hard drive is using?

It is generally recommended that you use a disk 2 - 3 times the size of your hard drive - the reason: TM does not delete backups; it creates an initial full backup and then keeps adding incremental backups of new content every hour. It will do so until the disk is almost filled; it will then delete the oldest backup to make room for the latest backup. It is designed to have backups available for you from many dates/backups in case you are looking for a particular one.

It shouldn't. Your first backup will be roughly the size of your system (there are the files backed up and likely some system overhead). Subsequent backups will consume additional space

How large is your internal disk? That is what it uses - not the actual amount of data on it. It wouldn't matter if you only had 10 GB on it.