How do you have a different photo on lock screen to Home Screen iOS 16?

If I go to Settings > Wallpaper and tap Customize under the Home Screen, I can then select a different photo for the Lock Screen only. I have one photo on my Lock Screen and a different one on my Home Screen.

Also note that I first locked the phone, then unlocked and touched and held the Home Screen until I was able to select which Home Screen I wanted to customize. See the following from Personalize your iPhone Lock Screen - Apple Support

You can create multiple custom Lock Screens and switch between them throughout the day. If a Lock Screen links to a specific Focus, switching from that Lock Screen to another also switches your Focus.

1. Touch and hold the Lock Screen until the Customize button appears at the bottom of the screen.
2. If the Customize button doesn’t appear, touch and hold the Lock Screen again, then enter your passcode.
3. Swipe to the Lock Screen you want to use, then tap it.

After doing that I then went to Settings Wallpaper and tapped Customize under the Home Screen there.

Great.

I do think the whole thing looks a bit first generation and hopefully will eventually get some improvements.

Have you tried the following? Go to Settings > Wallpaper and tap Customize below the Lock Screen to change the Lock Screen wallpaper.