connecting Time Machine to hard drive

exFAT is not a supported format for Time Machine. Assuming that there is no data on the disk, you should use Disk Utility to reformat the drive to APFS format. Once you do, the Mac should allow you to set up Time Machine.

Be aware that formatting a disk will delete all existing data on the disk.

Erase and reformat a storage device in Disk Utility on Mac - Apple Support

Choose Mac OS Extended (Journaled) and give that a try.

Nice work.

Although you chose Mac OS Extended (Journaled), the drive was likely formatted in APFS (Case Sensitive). You don't have to worry about this, just an FYI.

To confirm, right-click on the drive icon, then choose Get Info and check the setting for Format.