From Finder unable to drag and drop into non-Apple external drive

You have omitted details about the external drive and about the other system—omissions which usually means you’re using Windows and Windows formats—and that with what details have been provided that likely means the storage device is formatted using NTFS, which means you can either erase and reformat the volume using a format supported for write by macOS or by both macOS Windows (e.g. ExFAT), or can install a third-party NTFS app with write support (e.g. Paragon), or you can ditch the external storage and set up a file share from your Windows system or set up a share from your Mac and can mount and access that share from the other system(s) via local network, or can install a Network Attached Storage (NAS and access that from all local systems.

A NAS with Time Machine support can also be used for Wi-Fi or wired Time Machine backups from all local Macs.

And to avoid possible confusion, your footer here indicates a MacBook Pro laptop, which is a very different system from a Mac Pro desktop.

what format have you used for the drive?