Steps to reproduce:

Disable Local DNS Caching.

Set eero’s DNS settings to whatever you want, in my case I choose https://www.dns0.eu/

On Ubuntu my /etc/resolv.conf looks like this:

nameserver 193.110.81.0
nameserver 185.253.5.0
nameserver 2a0f:fc80::
nameserver 2a0f:fc81::
search .

Output of https://github.com/macvk/dnsleaktest/blob/master/dnsleaktest.sh looks like this:

You use 1 DNS server:
2a00:11c0:e:ffff:1::1003 [Germany AS42473 ANEXIA Internetdienstleistungs GmbH]

All as expected to this point.

Now change /etc/resolv.conf to the IP of my eero gateway:

nameserver 192.168.4.1
search .

(Or enable Local DNS Caching - then DHCP pushes the gateway IP as DNS …)

Output of https://github.com/macvk/dnsleaktest/blob/master/dnsleaktest.sh looks like this:

You use 8 DNS servers:
2a00:11c0:e:ffff:1::1003 [Germany AS42473 ANEXIA Internetdienstleistungs GmbH]
88.134.228.33 [Germany AS3209 Vodafone GmbH]
88.134.228.97 [Germany AS3209 Vodafone GmbH]
172.253.11.5 [Switzerland, Swiss Confederation AS15169 Google LLC]
172.253.12.3 [Switzerland, Swiss Confederation AS15169 Google LLC]
172.253.14.3 [Switzerland, Swiss Confederation AS15169 Google LLC]
208.69.33.64 [Germany AS36692 Cisco OpenDNS LLC]
208.69.33.81 [Germany AS36692 Cisco OpenDNS LLC]

first is dns0, second and third are ISP DNS, then Google DNS and OpenDNS

only the first one is configured :/