The first time I ran into this behavior with local DNS servers I was also surprised at how convoluted such a basic setting is in HA (HAOS). Made me remember the same situation with NAT hairpinning on Cisco zone-based firewalls. A common setting with a standard, universally understood configuration is implemented in a unique and thus confusing way.
If I understand it correctly, in HAOS Home Assistant Core uses an internal DNS server that runs as a docker container: hassio_dns.
Network settings, including DNS server, can be changed on the web GUI > System > Network > Configure network interfaces. Or from the HA cli using ha network. DNS settings specifically (I think this is for hassio_dns) can be managed from the HA cli as well with ha dns.
On my setup a set the same local DNS server with both ha network and ha dns.
This is one of those pieces of the config that warrants a dedicated page in the docs, but I haven’t found it. Please point/correct me as needed.