I have a back up external drive "WD My Passport for Mac".

<< can I use this passport to also back up my apple mac Book pro or do I need to get another passport drive??? >>

in my opinion, you would be better off using one drive per computer.

However, if running 10.13 High Sierra or later, you can use a seldom-seen option of File Sharing to make that same drive available as a shared backup destination.

Back up to a shared folder with Time Machine on Mac - Apple Support

This option is complicated by a fundamental conflict over space available on the destination backup drive.

Drives like the MyBook came with software, and the manufacturer suggested you install that software.

What that software does is to "spare you the annoyance of having to re-initialize the drive" from its factory set Windows New Technology File System (NTFS) or similar unusual (on a Mac) format. Instead, the software they provided would simulate a MacOS drive inside an NTFS file for your Mac.

The downside of using the manufacturer's software is that if the manufacturer's software is not loaded, the Macintosh Volume is not readable. This means that in startup manager, Installer, and in Recovery such as after a data loss, the Macintosh Volume will not be visible.

The standard advice given here to all users, including novice users, is to discard the manufacturers software and NEVER use it. "Best Practice" is to erase the drive when new, using only MacOS Disk Utility, and create the fundamental data structures needed for consistent, reliable use by MacOS.

Be sure to "show all devices" which will allow you to ERASE the entire Device (physical Drive) by its immutable manufacturer-given device-name, not any user created Volume-name.

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Just open Disk Utility and erase the disk with with an APFS format with GUID partition map. To use with other another Mac, then just partition the disk using the Partition option in Disk Utility.