How can I read Exfat files with my Macbook M2 Chip?

Is macOS prompting you to allow the external drive to connect? With Apple Silicon Macs, users are not prompted to confirm it is safe to allow an external device to communicate with the computer & macOS. You can go into the "Privacy & Security" System Settings and temporarily change the setting "Allow accessories to connect" to "Automatically when unlocked" (found at the bottom half of the Privacy & Security Settings) while you are testing this external drive just in case you are missing the system prompt for it when connecting the drive.

Allow accessories to connect to Mac - Apple Support

Try booting into Safe Mode to see if the drive is seen & mounted. If this works, then you most likely have some third party software installed which is interfering with the normal operation of macOS.

Make sure to connect the drive directly to the laptop. Disconnect all other external devices in case one of them is causing a problem.

Does the external drive support using its own power supply? If not, then maybe try connecting the drive to a powered dock or hub in case the drive needs more power to function.

If you need to use an adapter, then try using another adapter especially a different brand. Try using another cable as well.

What is the exact make & model of the external drive?

Do you have access to any other computer you can use to verify this drive still works?

Verify that in the Finder Settings : General, you have the ability to show external disks on the Desktop.

Reboot the M2 Air and connect the exFAT drive again in a different USB-C port. If it fails to mount, then that exFAT device is toast, unless you can mount it on that 2018 MacBook Pro to extract the files to a FAT32 USB stick. If you can do that, I would boot that 2018 Machine into Recovery mode and then run Disk Utility First Aid on that exFAT device and again try to use it in the M2 Air.

"it is toast" = the drive has failed.