Relationship fails, who gets what?

The owner of the property including all structures is the one registered in the land register. Therefore, it is always advisable that both partners are registered as owners with a 1/2 share in the land register. Otherwise, there are the regulations of the equalization of accrued gains for married couples. Without marriage, only the first option offers effective protection.

I have not yet been to the notary for a consultation, but is it so that:

If the girlfriend / partner owns the property,

then both are the homeowners?

House and land are inseparably connected.

House and property are inseparably linked.

Can it then be registered in the land register that both are homeowners?

It is possible to register in the land register that 2, 3, 1000 people and with which share someone is the property owner. Whoever owns the property is also the owner of the buildings. Principle of property law. It was different only in the GDR era. Possibly you can have encumbrances such as right of residence, easement, etc. registered in Section 2 or 3. I cannot provide legal advice for free, please consult a notary.

I was just at the notary for a consultation.

So what I can say now:

1) Only the person who is registered as the property owner in the land register is the homeowner

2) Whether one is married or not makes no difference

I was just at the notary for a consultation.

So what I can say now:

1) Only the person who is registered as the property owner in the land register is the homeowner

2) Whether you are married or not does not make a difference

You can't say it like that. Whoever is registered in the land register is the owner under contract law.
From a civil law perspective, the situation is different.
Both are the beneficial owners. Accordingly, both have a claim to the building.
It is important to record exactly how much each has invested in the property (including documenting any personal contributions).
In the event of a separation, the person remaining in the house must compensate accordingly.
The value of the contributed land would, of course, be credited to the partner.

Regards