Can I transfer files by connecting an external drive with a Lightning port to my MacBook Air?

If TM is actually using the drive, then it was reformatted by TM for its own use. It's a feature. When TM asks if you want to use the drive for backup and you reply yes, then it will also reformat the drive at that time prior to making its first b/u of your stuff.

You can confirm the drive's format by right-clicking the drive icon on the desktop then select Get Info. You will not have to reformat the drive if it now shows APFS or Mac OS Extended (Journaled) formatting.

Yes, the external drive likely has a USB-C port. You would use a USB-C to USB-C cable to connect it to your Mac.

Once connected, the drive should be displayed and accessible on the Mac's desktop.

You may, however, need to format the drive for use with the Mac as most off the shelf external drives are pre-formatted to work with PCs.

Guidance for using external drive can be found here: Connect and use other storage devices with Mac - Apple Support

Guidance for erasing and formatting a drive can be found here: Erase and reformat a storage device in Disk Utility on Mac - Apple Support

Also, in Finder Settings you may need to enable "show external drives on the desktop".

FYI, if you will be using the drive for Time Machine back up, then it must be reformatted to either Mac OS Extended or to APFS. After which it can only be used by TM and not for incidental storage of other data as it will be write protected by TM until you remove it from TM use and reformat it again.

So have you now successfully connected the drive to the computer?

You're welcome.