Can we afford a home?

Then clearly: If you do the whole thing, then start saving. Restaurant visits €100 --> there should be a maximum of €10.
Clothing: €150??? Much too much...

Saving, fine and good, but you really have to be careful not to sacrifice too much quality of life in the process. If I no longer go to the Italian place around the corner or only wear last year’s KiK collection just so I can somehow afford a home of my own, life quickly takes on a sour taste. This years-long loss of quality of life won’t eventually be compensated by having your own four walls.
If you have the opportunity to get into your own four walls early, you should do it. Especially in the early years, it is often hardly possible income-wise to keep up with price developments.

Saving, all well and good, but you really have to be careful not to sacrifice too much quality of life in the process. If I no longer go to the Italian restaurant around the corner or only wear last year’s KiK collection just so I can somehow afford a home of my own, life quickly acquires a bitter aftertaste. Eventually, even your own four walls will no longer be able to compensate for this years-long loss of quality of life.
Those who have the opportunity to get into their own four walls early should do so. Especially in the younger years, it is often hardly possible income-wise to keep up with price developments.

living in the wine cellar but rarely drinking

We have savings of about 6k, which we have also planned for the notary fees.

Sorry, but I wouldn’t exactly call 6K a reserve. When building a house, that’s just the difference between whether or not I want a pattern in the tiles.

When did I mention that we have no reserves at all? We don’t have equity in the sense of 20-30k, which is often expected here. We have savings of about 6k, which we have also planned for the notary costs. It is really unpleasant that false facts are being presented here.

I am not presenting false facts. You did not mention this 6k in the initial post, you wrote: no equity. I also did not receive an answer to the question of how long you have been saving the mentioned 1200 euros per month. And honestly: it is nice that you are planning this money (your entire reserves or do you have more in reserve that you are not mentioning?) for the notary, but what are you paying the property transfer tax with? Please don’t be mad at me, but the project is currently not for you. Wait a few more months and save some equity first.

I would like to see my comment on the ownership question addressed again.

The construct is based on the assumption that both persons will also be borrowers. If he is not completely stupid, he will want 50% in the land register. That means the prospective parents-in-law also gift him (tax?), is everyone clear about that? What happens in case of separation? Can mom bear that the house then has to be sold and he will receive half the proceeds?
Are there any other heirs besides the OP who need to be paid out?

The OP writes in the profile that the place of residence is Münster. Not a cheap spot.
Either the house is worth significantly more than stated, which complicates the issue of gifting to a third party (her boyfriend), or the property is in poor condition. Then the 60k renovation costs can be questioned. Is there a cost breakdown for this? Who made it? Ever visited a bank with it?

In the case of a sale to a child, is there real estate transfer tax? Or only on half of the prospective son-in-law?

Nothing is really given as a gift if the estimated house value is correct. Surely one doesn't have to tighten the belt excessively to finance the desired amount with the stated income. According to the household calculation, the two are also relatively frugal. They are just at the beginning of their careers and have had a few investments (car, repayment of student loan). If there is no urgent rush from the parents, perhaps the two should still wait another year, save €20,000 (as there should now be €1,600 left per month), and approach the project with this buffer in the background. And, as has already been advised here several times, consider the emotional upheavals it might cause if the family home should be put up for disposition due to a possible separation. And how to deal with it.