Client name in Network Overview

That is correct. I have it turned off on my home network which allows Pi-hole to see me and identify an Apple device. I've left it on (default) for all other networks, and for those it will use non-identifying MACs.

Access the feature at Settings > Wi-Fi > tap the next to the network name > Private Wi-Fi Address

Note that once it is disabled for your your home network, the phone will display the phrase "Privacy Warning" just under the network name. If you tap the network name you'll see a text panel which advises that it is disabled, along with a handy explanation of the options. Nothing to worry about, since you trust your home network anyway.