Can we really afford this, and will the bank support it?

Offered does not mean that it will also go for that price. Price of the new property 1.6 million, i.e. around 100,000 EUR just for incidental purchase costs.

And you first have to get the new place. In those price ranges, I can imagine someone coming by and saying, "I want it, I'll give you 2 million if necessary."

Go to Bavaria in the situation of (large) cities. You will be surprised how little has happened there in recent years. For new and good properties in good locations, the price there still only knows the way up. There really is still a seller's market.
So yes: the well-earning Bavarians actually stand in line. Maybe the line is no longer as long as before and prices no longer rise 10% per year (but only 4%), but new buildings are in demand.

I suspect that the house sale was not planned far in advance but unfortunately there is probably a divorce etc. behind it.

Then I interpreted the statement about the "new building" incorrectly: I thought it was a projected new building.

Then I misunderstood the statement about the "new building": I thought it was a planned new building.

No, the house actually already exists. We will now simply arrange a viewing appointment. According to the owner, the sale is supposed to be completed by June. If you "click" and agree on details, we are allowed to go to the bank and report on our project.

I can already see a residual risk of finding a buyer for your own home just in time while the seller of the target property is hoping for a quick notary appointment including payment of the purchase price.

Are there such things as "bridge loans" through the financing institution in these situations after the value of the new house has been checked by the bank/appraiser?

Runs smoothly. Do it and enjoy the new house!

@nordanney Are there, after the value assessment of the new house by the bank/appraiser, such things as "bridge loans" from the financing institution for such constellations?

Sure. Interim financings. Variable or fixed. Repayment through the sales proceeds. But it will only work with the bank that is already financing.

You then have 50% of your income tied up in mortgage and incidental costs. That doesn't include a single cent for reserves or saving for something new.

I also don't understand how with 8,000 EUR net, a loan rate of currently 2,000 EUR, 3 cars, many trips, etc., you want to save 4,000 EUR per month. So vacations, 3 cars, often eating out only cost you 2,000 EUR/month?

I think that's wishful thinking.

I would let it be.