As stated, ideally I would be able to just make a backup image of my phone to an external harddrive (with or without a computer in the middle).
Phone breaks, just flash the image onto another same-model-phone and continue.
Or is this impossible, because things are registered for individual devices?
I agree, I wish it were this simple like with traditional computer management, but on mobile it isn’t that easy for several reasons. I would suggest that a starting point to see the challenge is due to encryption layers and not having root on the device. Both of these are by-design to limit threat vectors, but both contribute to it being not easy to do a full system backup and restore.
There are many many more pieces, such as the complications of the overlay filesystems, etc. that make it so you aren’t just looking at a simple partition structure. I am honestly not the best to answer this, I am a Linux guy so I can sympathize with you wanting to have full granular control, but concessions in control are made in the name of threat reduction.
Now, some ROMs such as https://lineage.microg.org and LineageOS itself use user-debug builds meaning that you have access to the underlying system using adb root. However, as policy, iodé only makes user builds which do not allow the use of adb root. You will notice that using the “Backup” utility in System Settings will not comprehensively backup all app files and settings for this reason, making it a bit of a hassle to migrate between phones (but the backup utility does assist with a lot such as getting the apps reinstalled albeit with a lot of them having their user data missing upon reinstall).
Others may contribute some more concise and yet more detailed replies, let’s see what other comments we get.
As long as Vince doesn’t decide to change something here, we can fantasize about it a thousand times more
Some years ago, I was successfully using TWRP (with LineageOS) which just allows to do that, but with my Fairphone 4, I had so many severe problems, that I have abandonded this way. Still, it would be nice to have a full dump of everything that is on the phone to an external drive, e.g. using the Linux program dd AND being able to copying the whole thing back to the phone..
Again, I do sympathize with this view, and personally would welcome it. But from a security / privacy perspective if you can do this, then what is stopping a malicious actor / privacy invading 3rd party (sadly like a government agency like in my home country) from doing the same?
Doing this from recovery directly (TWRP or other) I don’t see this as ever being possible, since at the recovery level you haven’t yet unlocked your userdata partition. If you can read the data at that level it is effectively bypassing encryption (see malicious actor above).
But but regarding using adb root which would need interaction from the user on an unlocked phone to connect, while I haven’t seen credible evidence that a userdebug build (that allows adb root) could be compromised this way, I think that is the theoretical reason that it is considered less secure and not enabled. And as @AlphaElwedritsch has mentioned here and many places, “until the decide to change something here, we can only fantasize about it” 
I also am still not certain, maybe @AlphaElwedritsch or @petefoth can clarify: can adb root be used on a device that has a re-locked bootloader?
I don’t know: none of the devices I own support bootloader locking (and its is not somethig I would do even if they did (see here and here). However I think that rooted debugging (adb root) and bootloader lock/unlock are not really related, so I would expect that, if the ROM supports rooted debugging (i.e. it is not a user build), then rooted debuggin should be available even if the bootloader is relocked. I would be interested to know the answer, and I may go digging elsewhere to see if I can find an answer. I’ll report back
From the sfX_Android Matrix room / Telegram Channel
yes you can re-lock on userdebug on supported devices (afaik)
I see no reason why adb root should have any effect on it as long as you do not tamper with the device (depends on if there is verity enabled etc etc)
The whole backup topic is really sad, isn’t it?
As mentions before e.g.: Real Backup needed - #2 by AlphaElwedritsch the only “solution” today is GitHub - seedvault-app/seedvault: A backup application for the Android Open Source Project. that still has limitations and is primarily developed and maintained by the Calyx Institute for usage in CalyxOS ?
You can find Seedvault on iodé, via settings → system → backup.
Sorry, i know that Seedvault is also available on iodé 
But independent of the ROM: Seedvault has limitation.
True.
But for me, it worked good enough (learnt a few lessons when using it to switch from CalyxOS to iodé, but that is to be expected with any program).