My Imac can't mount a Western Digital Sata drive (2 TB).

WD makes several models of external drives. Some come with an independent power supply; others like the WD Passport are "bus-powered," meaning they must get all power, both for their electronics and turning the platter motor, from the computer's USB port. Computer USB ports have a finite amount of power available.

Yes, the drive could be feeling its age. Electric motors like the one which spins your external drive have a habit of demanding more power as they age and wear. If you have a bus-powered drive that is older, it could have reached the point of demanding more than some computers' USB ports can supply. Maybe the PC at the computer shop had different USB power specs than a Mac. BTW, a "Think Pad" is another PC, not a different Mac.

If you have the typical bus-powered WD Passport, understand that those are entry-level products not designed for performance or longevity, but rather to go on sale at the Office superstores every fortnight. I recommend upgraded to pro-grade drives. However, there are a few thing you can try to keep the WD is service longer (⚠️ note: these steps do not apply if your external drive has its own power supply):

• If you do NOT need to use the drive on a Windows computer, use Disk Utility to reformat the drive to Macintosh Extended. Sometimes that reduces the power demand.
• Try a "Y" USB cable it connects one drive to two of your USB ports to get extra power to the aging motor. See this. Make sure the "little" end looks like the connector on your external drive.
• Get a POWERED USB hub--has an independent power supply. This take most of the load off the computer's port and gets plenty of power the the drive.

I struggled with "name-brand" external drives for years and finally took the advice of wise people here to go with only pro-grade external drives. All we have now started as bare enclosures I bought from Other World Computing (OWC). I install bare WD "Black" 3.5-inch drives. None have even given a hint of trouble. The extra price became palatable whe I asked myself, "What's my data worth?"

Our iMac have this model: https://eshop.macsales.com/item/OWC/ME3UHKIT0GB/

Being a USB3-capable unit it will still work if you get a newer Mac later.

If the drive is formatted as NTFS— you may have to update your third party software.

Paragon

Tuxera

Do any other USB things work in that port?

What does System Report>Hardware>USB show?

It could be the case, since a Think Pad is also Windows, but for Windows to see it it seems it must be NTFS or ExFAT.

With it connected to the Mac open Terminal & paste this in & hit return...

diskutil list

diskutil list

/dev/disk0 (internal, physical):

#: TYPE NAME SIZE IDENTIFIER

0: GUID_partition_scheme *500.1 GB disk0

1: EFI EFI 209.7 MB disk0s1

2: Apple_HFS Macintosh HD 499.2 GB disk0s2

3: Apple_Boot Recovery HD 650.0 MB disk0s3

/dev/disk1 (external, physical):

#: TYPE NAME SIZE IDENTIFIER

0: GUID_partition_scheme *6.0 TB disk1

1: Apple_HFS 10.8 750.1 GB disk1s1

2: Apple_HFS 10.13 605.7 GB disk1s2

3: Apple_Boot Recovery HD 650.0 MB disk1s3

4: Apple_HFS 1012 748.3 GB disk1s4

5: Apple_Boot 650.0 MB disk1s5

6: Apple_HFS 10.10 749.6 GB disk1s6

7: Apple_HFS 10.9 749.6 GB disk1s7

8: Apple_Boot Recovery HD 650.0 MB disk1s8

9: Apple_HFS 10.11 750.1 GB disk1s9

10: Apple_HFS 10.7 750.1 GB disk1s10

11: Apple_HFS 10.6 750.1 GB disk1s11

12: EFI EFI 209.7 MB disk1s12

How exactly is it connected to this Mac?

So no hint of a drive in the USB part?

Do you know what format it is?

As KiltedTim suggests... the drive may be or has failed, perhaps the enclosure.

No APFS on the internal drive is fine...

Types of disks you can use with Time Machine on Mac

You can use Time Machine with a Time Capsule, and with USB, FireWire, and Thunderbolt disks. The disk can be directly connected to your Mac or be on a network. If a disk has partitions, you can use one of the partitions for your backup disk.

Time Machine can’t back up to iPhone, iPad, or iPod or to a disk formatted for Windows. If you connect a disk formatted for Windows, it can be reformatted (permanently removing all data) to a Mac format and used as a backup disk.

The most common format for a Time Machine backup disk is Mac OS Extended (Journaled) format, but Time Machine also supports Mac OS Extended (Case-sensitive, Journaled) and Xsan formats.

Important: You can back up from an HFS+ or APFS-formatted disk to an HFS+ disk; however, Time Machine can’t back up to an APFS-formatted disk. If you select a back up disk formatted as APFS, Time Machine offers to reformat it as HFS+.

If the disk uses the Master Boot Record (MBR) partition type, some partitions may not be available for use with Time Machine.

If your backup disk is on a network, the network server can use Server Message Block (SMB) file sharing. Your Mac must be connected to the SMB server when you set up Time Machine. After you select the network disk in Time Machine preferences, Time Machine automatically connects to the disk when it’s time to back up or restore your data.

Note: Some SMB and AFP disks from manufacturers other than Apple don’t support Time Machine. If your network uses an SMB or AFP disk, but the disk doesn’t appear in the list of devices available for Time Machine backups, contact the disk’s manufacturer.

https://support.apple.com/guide/mac-help/types-of-disks-you-can-use-with-time-machine-mh15139/mac

Time Machine Can’t Use APFS Directly

APFS doesn’t support hard links to directories, and Time Machine depends upon them. These are directory entries that associate a name with a directory or file on a system. The term usually applies to file systems that allow more than one hard link for the same file or directory. In the case of APFS, your Mac automatically converts any directory hard links you may have to symbolic links as you transition to the new file system.

This has a huge impact on Time Machine, which depends upon these directory hard links to work its magic. When Time Machine makes its first backup, it copies everything and then makes a dated folder, where it places hard links to the files it’s already backed up. The second backup uses more hard links for unchanged files, copying new or modified files into the dated folder for that second and each successive backup.

The fact that Apple’s new file system doesn’t support hard links to directories means you can’t use an APFS-formatted disk as your Time Machine backup drive. If you attempt it, Time Machine will automatically convert the volume back to HFS+ and use the drive that way.

https://support.apple.com/en-gb/HT208018

Have you considered the possibility that the drive has failed?

Nothing lasts forever. Mechanical hard drives have moving parts. Moving parts wear out. Eventually, they fail.

The electronics in the drive may have failed as well.

Try plugging it into another computer. Even if it's a Windows machine that can't read the Mac formatted drive, it should at least know that something is connected...

That’s a great explanation. I have a bunch of OWC enclosures, going back years (why do I still have an ATA enclosure limited to drives smaller than 128GB? Help!).

However, I have found that even some of the dirt-cheap ones from Sabrent or Inateck are far easier to work with than the sealed-up cheap cases used by WD, Seagate, and others. They are made to be easy to open and work with. To install drives in some of them, you only need to have thumbs. Cases for 3.5” drives generally require a screwdriver in addition to thumbs. I keep a few spares on hand for 2.5” and 3.5” drives. I also keep a supply of spare cables.

The suspect drive mounting in Windows points to a different problem, though, as mentioned.