Hi @Timo_Rohrig ,
Let me add some important information regarding SEO here.
It is really NOT recommended to use 2 different domains for the same website. You can have more than one domain pointing to a website but you really need to choose only one as the default. So, any other domain should make a redirect (DNS redirect preferred) to the main domain.
Google will probably not index your second domain even after you add the sitemap. Imagine how confusing would be for users searching for you content. And, imagine how easy would be to have higher rankings on SEO if you could have dozens of different domains ranking well on Google.
So, it may be the case that Google will just choose the domain it will list. But it can get worse, your 2 domains can get penalized for "duplicated content" and you may have problems to find them both on Google.
So, looking very quickly to your website and trying to understand what you were doing I would suggest 2 options for you:
Make 2 separately websites, one for physiotherapie and the other one for osteopathie. You would not need the homepage with the 2 options but you could still have one siting linking to the other one. This strategy may help you a little bit with SEO in the short term, but I wouldn't recommend it in the long term.
Choose a different new domain, like seitz-jaeger.de (or any variation) and keep the current website structure. I am assuming here that Seitz and Jaeger are names. If I have understood everything correctly that would definitely be my recommendation. I believe you won't have problems to rank well for both terms (physiotherapie and osteopathie) if you create good content for these terms, regardless the domain you choose. You could still keep these 2 domains pointing to the new one as a DNS redirect, but they would not make any difference in terms of SEO.
Let me know if there was any misunderstanding here and I can try to help you more.