Does the concept of Enterprise Keywords exist in SharePoint Online/2016?
Yes. It is available.
You can add an enterprise keywords column to a list or library:
For SharePoint_Online, click Settings Settings icon and then click Library settings or List settings.
In the Permissions and Management column, click Enterprise Metadata and Keyword Settings.
On the settings page, in the Add Enterprise Keywords section, select the check box for Add an Enterprise Keywords column to this list and enable Keyword Synchronization.
Yes it is. Enterprise keywords are pat of Managed metadata service application. For on prem you have to make sure it is configured properly.
You can learn more about it over here: Add an enterprise keywords column to a list or library