Syncing your GNOME Extensions is done via a Chrome/Chromium extension.
Once installed, you should see an extension with the GNOME logo as its icon.
Left-clicking the icon will take you to the GNOME extension site.
Right-clicking the icon will bring up a context menu
Right-click the icon, and select "Options"
- A menu will pop up with one element (fourth from top as of time of writing) that will be for enabling extension syc:
Synchronize GNOME Shell extensions list: If enabled, your GNOME Shell extensions list will be synchronized with your Google account.