Can the document desktop stacks be split for a specific file extension? I have documents I would like to have stacked
on my desktop. 

Specifically, to group all files with a particular extension all together using Mojave's new Stacks feature. 

Is there a way to register a new stacks kind?
No - Apple designed the default configuration but does allow you to stack on tags, which are user definable.

You could also make a smart folder to locate all the specific files you want, and then curate the tags easily, but not automatically out of the box.

Source:

Here's how to use tags for custom desktop Finder stacks.

In the image below, you can see the option under Group Stacks by -&gt; Tags. 

Each of the files you want to put in this Stack must have the same Tag.  When you do this, macOS will create a custom Stack for you for all files that contain that Tag.

This is the only way that I'm aware to create a custom Stack.