I do not use Timeshift, but I have a thought on your question.
Maintain two folders for Timeshift naming them “laptop” and “Geekom”. When you are using Timeshift on the laptop, rename the folder “laptop” to “timeshift”. When you are using Timeshift on the Geekom, rename the folder “Geekom” to “timeshift”.
You could probably also use partitions each with its own folder for “timeshift”, but this would probably be less flexible.
What about splitting the hard drive in 2 EXT4 partitions? 3tb + 3tb? So in one you can manage backups from one machine, on the second partition for the other machine. What do you think about? I’m pretty sure it would work flawless.
@ricky89 Provided a great solution. It’s what I’ve done. I have a 500GB SSD removable drive I use for regular TimeShift backups for seven different systems. I’ve partitioned the drive with seven different partitions, one for each system. I just plug it into a system, TimeShift knows which partition it is set up to use for that system, and it just works. Pretty nifty.
Be aware, however, that SSD are not the best solution for long-term backup. They can die over time if not regularly connected to power. But for regular TimeShift backups (that get regularly replaced) an SSD works great.
Did you have any troubles with this partitions encryption (luks)? Or you didn’t try?
I have not tried with encryption but I cannot see any reason it would not work. Encryption occurs at the block device level so I would think it should be fine.
Long-term is how long? Several years? Can you recommend something better?
Magnetic media (e.g. “hard” drives) is still the best solution for long term storage without requiring a power source.
Any NAND flash device (SSD, NVMe, USB drive, etc.) is fine for interim storage of data and can be used long term as long as it is either constantly or, at least periodically, connected to power. Over time, without power, they will lose integrity and there is the chance for data loss. However, even with power, over long periods of time (most research indicates ten years or so), they can still degrade and replacement should be considered at some point after to avoid data loss.