My new MacBook Air M4 won't recognize external Toshiba Canvio Flex HDD

macOS changed the FAT/exFAT driver beginning with macOS 14.x Sonoma from a Kernel based driver to a user space driver...perhaps that has some bearing. I would suggest trying to use Disk Utility to erase the Toshiba drive using your M4 to see if that makes any difference. You may first need to erase the whole physical drive using one of your older Macs (erase it with anything other FAT/exFAT just to be safe). See the instructions in the following Apple article for erasing the whole physical drive:

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

You can also try using HFS+ or APFS for the file system instead of exFAT to compare results to gather more information while troubleshooting.

You can also try using another USB-C cable as well (make sure it is a good quality cable).

Are you connecting this drive directly into the computer? If so, then try connecting the drive with a USB3 hub instead. I've seen some people report some drives won't work properly unless used with a hub. If you are using any docks/hubs, then connect the drive directly into the computer.

What happens if you already have Disk Utility open before you connect this Toshiba drive?

Try changing the Privacy & Security System Setting "Allow Accessories to Connect" to "Automatically when unlocked". This ensures that this setting is not preventing access to the physical drive....many times the system won't remember you allowed the device, or you may not see the pop-up prompt to allow the device to connect. At least do this while troubleshooting this external drive issue.

Try updating macOS to the newest version 15.4 which was released a few days ago.

I had a somewhat similar issue with a Garmin handheld GPS unit (formatted under FAT32) that refused to mount properly. Cables, adapters, etc were tested extensively and the unit mounted just fine on my 13" late 2011 MacBook Pro, two different Windows 10 PCs and my USB-C iPads (with proper adapters). The issue happened when GPS unit was connected with a microSD card inserted (it uses composite drives..internal GPS unit memory with an internal microSD card). The GPS would mount fine with the microSD card out of the unit but failed to mount BOTH drives when a microSD card was inserted.

This is what I came up with as a solution (after much head banging)...seems simple but it took me hours to figure out what exactly I did and in what order I did them in when I once got it to work as intended.

1. Log out of the current MacBook user account (stay at the login screen)

2. Power on the eTrex GPS

3. Plug in the USB cable (still at the login screen)

4. Wait a minute to a minute and a half

5. Login to the MacBook

To clarify, the steps above are no longer needed (at least in my case) since after updating macOS to the current version 15.4 the issue was resolved. In this case the fault was entirely with macOS.

I forgot to mention the model of the MacBook with the issue...13" MacBook Air (M2, 2022)