I think your mother-in-law network runs a DNS server on the router which does the resolving for you. So it's DNS, not mDNS. This is confirmed by failure to resolve the hostname when your phone uses private DNS. Your router, on another hand, doesn't resolve . local domain as belonging to local network.

When the devices obtain the IP address via DHCP from the router, the latter can specify the DNS servers to use. Yours may specify the provider's servers while mother-in-law's specifies its own.