That's most likely because... you didn't read the manual! :D
FallbackDNS=
A space-separated list of IPv4 and IPv6 addresses to use as the fallback DNS servers. Please see DNS= for acceptable format of addresses. Any per-link DNS servers obtained from systemd-networkd.service(8) take precedence over this setting, as do any servers set via DNS= above or /etc/resolv.conf.
Assuming your network is DHCP, edit your config eg. /etc/systemd/network/10-eth0.network:
[DHCPv4]
UseNTP=no
UseDNS=no
UseHostname=no
Your system will not pick NTP and DNS servers and also ignore the hostname provided by the router. Also make sure you ln -sf /run/systemd/resolve/resolv.conf /etc/resolv.conf