Error on my Mac: Backup not completed. Disk "Toshiba Ext" contains backups created on a different computer

Does that disk appear in Disk Utility?

Disk Utility User Guide for Mac - Apple Support

If it does, reformat it using APFS: Erase and reformat a storage device in Disk Utility on Mac - Apple Support

The short answer is to obtain a different backup drive, or to erase the one you have. If you decide upon the latter, did you choose it as the source from which to migrate your older Mac's information?

The obvious implication is don't erase the backup drive until you are certain you no longer need it.

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What specifics should i look for in the new external drive?

The answer is "it depends" — for example, if you intend to use it exclusively for Time Machine, then speed is utterly irrelevant. Simplicity and redundancy are far more important. Get any inexpensive, generic USB hard disk drive. Ideally, get two or more of them since any device can fail at any time for any reason, and you never want to be without at least one backup. That's the reason "inexpensive" is ideal.

On the other hand if you intend to use it for your Photos library, or for video editing, or anything that requires fast data throughput, get a fast SSD. Of course they are more expensive. Another thing to bear in mind is that when they fail, SSDs tend to fail catastrophically with no warning. Conventional hard disk drives almost always give you some warning, often becoming intolerably slow, gradually at first, before they become unusable.

If you don't know where to start: https://eshop.macsales.com/shop/external-storage