So according to this my mistake is using the Pihole as WAN DNS. I should use it only on the LAN section of the router. I will give this a go and let you know if it works.
L.E.: Everything seems to be working now.
Here is the current configuration:
- LAN DNS in the DHCP section of the router = Pihole VM IP address (IPv4)
- WAN DNS is the one from Google (8.8.8.8) with Cloudflare (1.1.1.1) as a backup
- IPv6 section = Pihole VM IP address (IPv6)
The only weird thing is that for some reason on my phone I see ads although I do not see them on my PC.
Later Edit: This post which helped figure out the fact that my Asus router insists on advertising itself as a DNS to the LAN devices in addition to the DNS settings explicitly filled in by the user. This helped figure out why my phone still sees ads (because it's using the router's IP as DNS for IPv6 and modern devices prefer IPv6 over IPv4).
The solution for the last remaining issue as far as I found out reading on Google, can be solved by using a third party firmware for my router (dd-wrt or Merlin) but considering that for the moment my router is still in the warranty period I am not sure if I should go there yet.