Apple Wireless Keyboard A1314 tilde key
I'm not 100% clear on what you're asking, but what I can tell you is:


I use a number of Apple Wireless A1314 keyboards with various iMacs and MacBook Pros and the tilde ~ key works fine on all of them. 
I haven't actually used any of this model keyboard on a Mac running macOS Sierra, but I have on Macs running Mac OS X Snow Leopard all the way up to Mac OS X El Capitan and the ~ key works fine on all of them.
If you actually meant the ` key (which on my keyboard is the same physical key as the tilde ~ key), then this also works on all of them.
As for the Section Sign (§), I've always achieved that with a option6 keyboard shortcut (i.e. there is no physical key labelled as § on any of my Apple Wireless A1314 keyboards.


If the above isn't helpful, you could try starting your Mac in Safe Mode just to see if that makes any difference in your scenario.

Boot into Safe Mode

Follow these steps boot your Mac into Safe Mode:


Fully shut down your Mac
Restart your Mac
Immediately press the Shift key and keep it down 
Let go of the Shift key when you see the login window (NOTE: If you have FileVault enabled you may need to log in twice).
Take a note of whether your keyboard now works as you want?
Exit Safe Mode by restart your Mac as normal
Test your keyboard again


Once you've tested in Safe Mode and afterwards, let us know how you went.