While I don't particularly trust Google all that much anymore, the fact that ISPs even have an opinion on this is a smoking gun that they're doing sketchy things with DNS data. There is no actual technical reason why they should care if you use their DNS servers or something else, even a private, encrypted DNS service.
Here in the UK sky recently pushed out an update to their firmware which made it impossible to use another DNS server other than theirs. There was a decent amount of push back, I managed to get them to downgrade my firmware but who knows how long that will last before they "accidently" update it again.
I also use my own router so I assume it doesn't affect me or does this mean that their network doesn't allow other DNS servers?
How would that affect a VPN? I use PIA and they have their own DNS servers.
I'll need to experiment with this when I get home I think.