Time Machine not showing partition/external drive as available disk

Hi,

I'm trying to make a redundant backup of my Mac on 2 different drives. One of them is the one I've been using for the last months now, which is working perfectly fine. The other one is an old drive which still contains an old backup from my previous Mac (the partition in the drive is called LACIE MBP).

The issue is that if I click on 'Select Disk...' in Time Machine preferences, the correct partition (LACIE MBP) in which I want to do the backup doesn't show up. Another partition on this same drive does however show up.

Here are some screenshots:

Time Machine Preferences screenshot. The 'LACIE OLD' is another partition on the same drive. I'm looking for the 'LACIE MBP' partition.

In the disk utility window, the disk just shows up as a Mac OS Extended disk (Journaled). It does however still recognises it as a Backup Disk since I used this for backing up my previous Mac as already explained.

I did already try to run the First Aid. Doesn't work.

Something else I noticed is that if I option-click on the Time Machine icon and click on 'Browse Other Backup Disks', I am able to browse through the old backup which is present in LACIE MBP.

Does anyone knows a solution for this, without reformatting the partition? (I don't want to lose the old backup which is still present on Lacie MBP)

Thanks!