I have two VPNs. One for privacy (e.g. general browsing) and one to reach my home NAS (e.g. to reach my selfhosted image backups). Is there a way to set the system up so that when I use applications X, Y and Z, it should use the privacy one, and if I use A, B or C, it should use the private one?
You could connect to your privacy VPN from your home server, then set up your home VPN to route all external traffic through the privacy VPN. From your phone, your home VPN is now both your home and privacy VPN.
Trying to do it on a per-app basis, you will very likely end up with whichever app is in the background trying to use the wrong connection.