I found the answer myself. It was indeed the wrong .htaccess configuration which was recommended by Hetzner.
The following configuration works:
DirectoryIndex disabled
RewriteEngine On
RewriteRule ^(.*)$ http://localhost:3000/$1 [P,L]
RewriteRule ^$ http://localhost:3000/ [P,L]
https://html.developreference.com/article/10474320/How+to+serve+Express+node+API+to+subdomain%3F