As long as you trust both your chosen DNS resolver and the VPN provider you are using, I don’t think there is any significant loss of privacy by using a separate DNS.
You will lose some anonymity since the DNS resolver you use will likely be unique compared to other people with the same IP address provided by the VPN, but in my opinion the anonymity provided by VPNs is so fragile that this is not worth worrying about much. If you’re being tracked via your DNS resolver, you’re likely being tracked by other fingerprinting methods that VPNs don’t protect against anyway. It is something to take into consideration though.
On Android, the OS custom secure DNS settings will override the DNS settings of the VPN you are using, but this varies based on OS. In principle it shouldn’t make a difference whether you set the DNS resolver via the VPN app or the OS itself, just be sure you understand which takes priority (keep in mind that your browser can be used to set custom DNS resolvers as well).