Your router always knows your Mac address, no matter how you got your ip assigned. And yes, you can use it to identify the client - that is why it is there. This whole post is nonsense written by someone who doesn’t really understand what dhcp is or how it works. Long story short, don’t look for privacy on local Ethernet segment :D
that doesn’t seem to be uniform behaviour. but i think we agree on the merit. if you are this paranoid, you just don’t use networks where you don’t have control over the local segment.
[admin@MikroTik] > ip arp print Flags: X - disabled, I - invalid, H - DHCP, D - dynamic, P - published, C - complete # ADDRESS MAC-ADDRESS INTERFACE 0 DC 192.168.88.160 A2:35:xx:xx:xx:xx bridge 1 DC 192.168.88.159 F4:60:xx:xx:xx:xx bridge 2 DC 192.168.0.1 44:32:xx:xx:xx:xx ether1 3 DC 192.168.88.168 18:3D:xx:xx:xx:xx bridge 4 DC 192.168.88.156 70:BB:xx:xx:xx:xx bridge