Construction costs are currently skyrocketing

We are straying from the topic, because it is about house building. And that should be regarded as a luxury and not as something given.

And you are wrong there. Living is something different than a sports car or a yacht. Construction prices, purchase prices, and rental prices are not three completely separate things; they are connected. When construction and purchase prices rise exorbitantly, this also has an impact on rents in the medium term. And that concerns everyone, because everyone has to live somewhere. More than that, a well-functioning society should enable everyone to live well: in an apartment that does not consume huge sums for heating, in an area where it is quiet and safe at night and where there are no shards of glass on the playground. Ownership for everyone is not necessarily required to achieve this, but ownership protects to a certain extent against extreme price increases in the housing market. As I said, in the medium term this is not decoupled. A landlord does not rent out of charity but to earn something with their money. And even if the apartment is already paid off, renovations are eventually due, and then construction costs also affect the apartment.

This 5000 Euro limit honestly surprises me because it doesn't seem particularly high to me personally. With our household income, we land pretty much exactly at this value, and I rather have the impression that we are more in the middle range in our environment.

I don't know your family situation but a couple with one child and 5000 Euro household income: 23% in this constellation are richer, 77% poorer. And with two children, it's 36% vs 64%. And with DINKS 13% vs. 83%.

Does anyone know good statistics on this distribution related to different regions? And this presumably always refers to households?!
I assume that the home ownership rate in rural areas is significantly higher than in larger cities or even university towns, for example.
At least it has felt like home ownership has been the norm in rural areas since my childhood.

Yes, it refers to households. You can relatively easily find a breakdown by federal states (destatis or simply Wikipedia) if you google for it. Yes, the city-states are very, very low, and generally the ownership rate in the East is also very low (which is probably due to expropriations and the general "no ownership" policy during GDR times?). In the western large states, we are still below the European average, but not as far off anymore.
(from Wikipedia "home ownership rate," with Europe and Germany average for comparison)

I don't know your family situation but a couple with one child and a household income of 5000 euros: 23% in this constellation are richer, 77% poorer.

And with two children it's 36% vs. 64%.

And with DINKS 13% vs. 83%

Hintertupfingen or Munich? The statistics don't matter, but it is decisive.

Hintertupfingen or Munich? The statistics don't care, but it is decisive.

Nationwide average. Words words words

Honestly, this 5000 euro limit surprises me because it doesn't seem particularly high to me personally. Our household income is pretty much exactly at this level, and I rather have the impression that we are rather in the middle range in our environment.

5000€ is already very generous. The problem is that with rapidly rising inflation it is worth less and less, and for example here in the house building forum you almost only deal with people who have a significantly higher income than average. Ask in the panel building complex who has a household income of over 3000€. There you might have that at most gross. They then say they don't know anyone who has a household income of 5000€.
Edit: According to Statista, from a net household income of 4500 and more, you belong to the top 10% income earners in Germany in 2022.