The Adguard DNS is for 5 devices. The Adguard app is for 3 devices. Do I need two subscriptions or is one included in the other? I’m on trial of the Adguard DNS, using it on my router, but my Android which has trial of the Adguard app does not show up on the Adguard main site as a connected device 
Therefore I consider using Adguard DNS to get config of all devices online.
Could Adguard DNS also include the Android app? Also, does the Adguard app’s “VPN” override the system DNS settings?
one included in the other
No, only the AdGuard VPN subscription includes AdGuard DNS premium.
my Android which has trial of the Adguard app does not show up on the Adguard main site
You’d have to configure the upstream in your AdGuard app to use your unique AdGuard DNS address, preferably its DoH/DoQ address as a new device, not sharing with the router.
Adguard app’s “VPN” override the system DNS settings
If Private DNS is active you’ll see that the DNS requests aren’t seen by the AdGuard app, on the other hand, Private DNS and AdGuard app DNS upstream setting will override the DHCP DNS setting.