How can I sync the Google keyboard's clipboard with Android's clipboard?

The actions in these answer are from a Galaxy Note 8 which is nearly identical to your device. Unfortunately I do not know any way to sync them and as far as I know, it is not possible. However, I can explain this behavior.

The clipboard is a feature of the keyboard which is nothing but an application that registers itself as an Input Method with the android operating system.

What you are referring to as the android clipboard is actually the clipboard from the Samsung keyboard. This keyboard is better integrated with the Samsung environment. Here's how you can switch to the clipboard from the Samsung keyboard:

1. Long Press on the globe like button:

2. Click on the Samsung keyboard in the list that opens:

3. Find the clipboard icon from the options on the keyboard, possibly in the extended list that is visible upon clicking on the 3 dots:

Particularly in certain text boxes like the email compose section, this clipboard is disabled for some reason:

Which is why you cannot use this clipboard everywhere even when you longpress in the area to bring up the clipboard shortcut.

To switch back to Google's keyboard, you can bring down the notification area and click on the change keyboard button:

However; since these are 2 different applications; there seems to be no way to synchronize them since the clipboard feature is probably not universally standardized.