Is a single-family house possible for 180k €?

Yes, it is still possible. It depends on the region. No idea what it looks like in NRW.

But your relationship also has to be strong enough for such a project.

Other relationships have already broken up while choosing wallpaper and tiles.
How much more so with your plan.

Building on your own takes at least 1 year.
That means a lot of stress – alongside work and private life.

Many things will take longer, cost more, not work right away, or not work at all – than expected.

Important topics are also:

Provision interest
(external) construction management
Insurance (keyword helpers)
Lump-sum contracts
Prefabricated garage (cheaper than built)
etc.

And you have to stay below standard – to be able to afford something "special" sometimes (e.g., electric blinds everywhere).

Due to the current construction boom, it will also be hard for you to get craftsmen.

I see it in my circle of friends. It is possible. But he is also single and only on the construction site.
Because a lot of soil had to be excavated and compacted with gravel, the buffer was already gone.

Building on your own takes at least 1 year.

I would plan for two years. When I think that our purchase from the developer already took one year (from signing to moving in) plus weeks before the signing and weeks afterwards (remaining work, etc.)...

Yes, with 2 years you really have to expect it, that's why "at least".

Also take a look at banks/insurance companies that transfer money based on construction progress/photo.

For example, the shell construction is done - photo - get 70,000 paid out. You don't have the hassle of submitting invoices.

And you can also immediately pay your helpers (without invoices), who of course do everything for free.