Thank you for your answers. I have now spoken again with the surveyor, who could explain it quite well.
There are two different issues at play here. On the one hand, there are the building regulations from the state building code (9m boundary construction, exempt from approval up to 3m ridge height and <45m2). And then there is the entry in the land registry. You can save money on this point. Especially because the surveyor in Brandenburg only has to come once and that after the completion of the foundation slab.
The entry in the land registry is billed according to the fee schedule and depending on the cost of the requested building project. There are rates up to 250,000 and from 250,000 €. So if I register the house and garage at the same time, I save this base fee of about 360 €, which is incurred if the garage is registered later and then calculated according to the table up to 250,000 €.
Conclusion: even if the garage is exempt from approval, costs for the entry apply. You can save if you do it together. However, I now have to have the foundation slab for the garage completed at the same time.
Regards
PS: Unfortunately, the garage has to remain 5 x 6 m. It will be a single garage. I want to be able to walk through between the house and the garage.