I transferred all my servers to a different netcup account.
This is what I learned, so you don't make the same mistakes that I did.

Hi, I'm Steffo and I have multiple servers (and domains) at netcup. This blog you're reading right now is hosted on a root server by netcup, on a domain I got through them. I think you get the idea.
The thing is... I had to switch my netcup account. I won't go into detail of why I had to do it, but I had to. Asking the netcup support, hoping for a way to simply change the details of the account, just gave me a "use the transfer tool" as an answer. So... I did that! In this blog post, I'll talk about how I did it, what mistakes I did on my way and what you can learn from my mistakes.
Step 1: Creating a new account
To transfer stuff to another account, you first need that new account. Makes sense, right? Well... On netcup, you need to order something to be able to create an account. Luckily, netcup has a free "Customer Account Registration" product on their page.
With this, you can go through the default account creation process. (Entering your data, waiting for netcup to create your account, verifying your ID, ...)
I can recommend not doing that stuff on the weekend, especially at the end of the day - you might have to wait until the next day until you get your account credentials and until your free order is processed.
Step 2: Backups??
Netcup says in almost every support wiki that all data will be transferred. Of course, I won't listen to some simple documentation, and I'll still panic about possibly losing data. Thus, I made backups of almost everything I have. All my DNS settings, docker container data, nginx settings, ... everything.
I mean, I wouldn't wanna lose all my stuff in the unlikely (impossible) event that everything is deleted after all.

Step 3: Error.
Now that everything is backed up, I can finally start the transfer. On the original account, I'm putting my servers into transfer-mode, which gives me a code that I have to enter on the other account. On the other account I enter the code and click on "validate", only to get an error back:
Please settle your unpaid invoices first. As soon as these have been booked and marked as paid, you can carry out the change of owner.
~ netcup Customer Control Panel
What?! When I look into my invoices tab, there's nothing. NOTHING.
Do you remember when I said, "until your free order is processed"? Yeah, that didn't happen yet. I created the account on Saturday evening and tried it on Sunday noon. I guess the verification or something else didn't finish yet. This meant that backing everything up and shutting everything down was useless. Whee.
"Wait, shutting everything down?"
Yes, I shut everything down before backing stuff up because in my opinion it's better to have a higher downtime than having to roll back (user) data (e.g. Matrix stuff).
This is how I operate and how I like to do it.

Step 4: Transferring the servers
A day later (Monday) I backed everything up again and started the transfer. This time it worked!
... at least until three of my domains were stuck at "Waiting for a response from the registry". First, I thought that I'll have to wait until the registry says it's okay to transfer. Then, about 30 minutes later, I saw: The original account got three e-mails, each with a question if I really wanted to transfer the domain. (Interestingly, my .de domains didn't do that!)
After confirming those three domains, the new account also got three e-mails asking if I really wanted to receive these domains. Of course, these are still my domains after all.
Once this was done, my domains were all on the new account. The last thing I had to wait for was the web hosting, which was transferred a few minutes later as well.
Thus, the transfer was done and I can use my new account!

(And yes, all data was still there, who could have known...)