Finding your hardware address: Apple Watch

The Apple Watch Series 3 and newer may connect directly to a wireless network- However, in order to do so, it will need to be registered as a separate device.  (Note: Apple Watches that only connect to a paired iPhone do not need to be registered.) To locate the hardware address for your Apple Watch you will need to open the Apple Watch app on an iPhone:

>> Basic Wi-Fi Address

1. Open the Watch app.
2. Go to General > About
3. Scroll down to find the "Wi-Fi" Address
4. Once you have completed these steps, you can register the Apple Watch to the network by following the steps for Connecting a device that does not have a web browser.

>> Private Wi-Fi Address

1. Open the Settings app, then tap Wi-Fi.
2. Tap the name of the network you joined. If you haven't joined the network yet, swipe left on its name and tap more .
   
3. Tap to turn Private Address on or off. 
   
4. Once you have completed these steps, you can register the Apple Watch to the network by following the steps for Connecting a device that does not have a web browser.

You will know that your watch is connected to the network if the wifi logo lights up on the watch.

Note

What is Private Wi-Fi Address? 

To improve privacy, watchOS 7 use a different MAC address for each Wi-Fi network. This unique, static MAC address is the device's private Wi-Fi address, used for that network only. The Private Address setting is on by default in watchOS 7. If the device is registered with private Wi-Fi address, it would potentially cause to re-register the device or cause other problems due to the change.

More info please refer to: Use private Wi-Fi addresses in watchOS 7