Cloned SSD, how can I clone it again?

I know that a system can't detect both SSDs at the same time because they appear as the exact same device, I tried to google it a bit but I don't know the real problem or description.

System can't effectively distinguish their filesystems in some cases because their UUIDs are cloned. Nonetheless, both devices are still fully accessible under /dev.

I cloned my SSD A to a similar SSD B with a live system and dd.

dd is rather inefficient at cloning - it will clone contents of unused disk space, which only wastes time and SSD write cycles. I'd suggest something like Clonezilla, which will only clone data.

Now that backup got old and I want to save SSD A again to SSD B.

That's a controversial backup strategy. On one hand, you can simply swap the drive and start working instantly. On the other hand, updating the backup is impossible, you're just creating a new one which is very inefficient and discourages frequent backups. During this process you have zero restorable backups and at all times you have one backup at most, with no way to rollback older versions or restore single files from them. Consider implementing a second layer of file-based incremental backups besides this one.