Have you ever used this Seagate drive before? If you previously used the Seagate software to manage the drive, then the software may have locked or encrypted the drive. You would then need to use the Seagate software to unlock, decrypt, or disable the security features before you can use it on the new computer.
If the Seagate drive is brand new it may be formatted as NTFS for use on Windows. While Macs can read an NTFS formatted drive, Macs are unable to write to them. If the drive is formatted with ExFAT, then perhaps it is using a block size incompatible with macOS.
If the Seagate drive is brand new, then just use Disk Utility to erase the physical drive as GUID partition and MacOS Extended (Journaled). If you need to share this drive with a Windows computer, then erase the drive with ExFAT instead of MacOS Extended (Journaled). Within Disk Utility you may need to click on "View" and select "Show all devices" before the physical drive will appear in the left pane of Disk Utility.
If the drive has been used previously, then connect it to your other computer and back up the contents of the drive so you can reformat (aka erase) it on the Mac using Disk Utility as I described previously. Then transfer the files back to the newly formatted/erased Seagate drive.
It is not necessary to use the Seagate software to use or manage the external Seagate drive. The proprietary drive software usually will just cause problems. It is better to just let macOS manage the external drive. If you need to encrypt the external drive, then let macOS use software encryption to encrypt the drive.
It is also possible you have a defective cable or adapter, or the Seagate drive could be defective/failing.