Does Time Machine automatically format an external drive for backup?

It may in some circumstances. See the following from Types of disks you can use with Time Machine on Mac - Apple Support

APFS or APFS Encrypted disks are the preferred format for a Time Machine backup disk. If you select a new backup disk that’s not already formatted as an APFS disk, you get the option to erase and reformat it. If the disk is a Mac OS Extended format disk that contains an existing Time Machine backup, you aren’t asked to erase and reformat the disk.

Also see the following from Back up your Mac with Time Machine - Apple Support

4 - Select your connected storage device and click Set Up Disk. During setup:

• Your Mac might ask if you want to erase the storage device so that it can be used for Time Machine. Either erase it or select a different backup disk or volume.

No. If a drive is not formatted in a way that the Mac understands, then the OS will ask if you want to format, eject or ignore the drive. If you choose to format it, then after it has been formatted, Time Machine will ask if you want to use it to back up the computer.

However the first time a blank drive formatted with HFS+ (Mac OS Extended) or APFS is connected to the computer, Time Machine will ask if you want to use it to back up the Mac.

EDIT - actually, @FoxFifth has posted a great response. “It may in some circumstances.“ 👍🏽