Replace sink with pre-wall element

Since it is apparently not clearly traceable, here is the chronology:

1. The apartment was equipped with a pre-wall system and a new washbasin during a renovation.
2. The tenants at that time subsequently removed the washbasin from the renovation and instead placed a different washbasin on a rickety cabinet.
3. The current tenant – me – took over the apartment as it was and removed the other washbasin along with the rickety cabinet because it seemed unsafe and unaesthetic. In the process, the condition of the pre-wall system behind it became apparent. The washbasin from the renovation is missing.

Well, what the previous tenant did can be totally irrelevant to you, at the beginning there was a sink on the wall, so you also insist on a sink on the wall. Definitely the landlord's matter...

No. At the beginning of his rental contract, it was just on a wobbly washbasin...

I would still report it to the landlord and wait to see what he says.

Hi,
then it’s still the landlord’s problem.
He could have demanded from the previous tenant to restore the original condition.
The deposit is partly intended for this.
If he didn’t demand it, it’s his problem and he has to ensure a properly fixed washbasin himself.

If the washbasin is simply ugly, however, it’s your problem if you accepted it that way when you moved in.

So, as I said:
If it’s not properly fixed - landlord’s issue.
What is/was with any previous tenants does not concern you.
Regards Olli

If he accepts the sink as it is in the handover protocol, is it the landlord's problem if he wants to install a different one? hmm, don't know

Clearly a landlord problem. Call directly, explain the problem, he has to take care of it. If he refuses, then reduce the rent.

I am grateful for any solution approach on how to mount a sink somewhat securely!

Hard to really tell from the pictures.

So, as I interpret it, you have to insert the part with the square side (T-slot nut) first into the profile and then rotate it 90° with the notches in the washer so it doesn’t fall straight out of the profile. Then insert a threaded rod and tighten it properly.
Normally, the threaded rod presses against the profile at the back and presses the T-slot nut against the profile, then it holds firmly. BUT, precisely at the point where the threaded rod should press, there is a recess in the profile. I think that’s the real problem.
Maybe there is space for a metal plate. Or shift everything a few cm within the profile.