Back up an external drive with Time Machine

Yup! So just to clarify, Time Machine excludes by default all external drives. Make sure to remove the drive from the exclusion list so that it will be included. And with a 2nd external hard drive (as your time Machine drive) you can routinely and automatically (without any intervention required) keep your internal drive + seagate drive backed up to the 2nd external whenever it's plugged it. You can leave it plugged in 100% of the time if you want, or only intermittently or occasionally plug it in.

Time Machine can only be used to back up macOS formatted drives, including externally attached USB drives with that one significant limitation. Is your Seagate USB mac-specific formatted, or is it something else like FAT32, ExFAT, or NTFS? I believe that would explain why it's grayed out.

Disk Utility is able to do two useful things that could apply to your situation. It can be used to copy all the files from one drive to another (like if you have an additional external drive you want to copy everything to), or it can also take your Seagate USB drive and compress all it's contents into a single backup disk image file which is then portable and good for archiving.