Start with the right checks.
A connected drive is not proof that the Mac is being backed up. A useful setup has an intended destination, a completed backup and someone who checks failures.
What to do next.
Choose storage appropriate to the Mac and your business or personal requirements.
On macOS Ventura 13 or later, open Apple menu > System Settings > General > Time Machine. Choose Add Backup Disk, select the intended connected drive, then Set Up Disk. Stop if an erase notice refers to files you still need.
Keep the Mac and destination available while the first backup runs.
Check the latest completed backup time. Include how the destination will be protected from loss, theft or accidental disconnection.
Check the result.
Restore a harmless sample to a separate location and open it. Confirm that the backup covers the information you actually need.
Stop and get help if…
Time Machine repeatedly fails, the destination holds the only other copy of files or backup coverage for cloud data is unclear. Discuss a broader recovery plan instead of assuming one drive covers everything.