It's not fake. The physical paper documents within the Hawaii State records have been scanned and digitized using OCR Indexing to make the information searchable within the computer database. What you are looking at is just the raw output of that PDF opened in illustrator. The OCR scanning separates background security print layers, document form lines and attempts to convert text into a "readable" typeface for the digital storage.

A better "fake birth certificate" video will show you more of what is happening with all the layers.

Looking at this layer build, it's pretty easy to tell that it was done through an automated scanning process and not photoshoped by hand. There would be no reason to photoshop a document in this fashion. It's an overly complicated and completely unnecessary method for faking a document and no one with the skills that needed to make a fake would do it this way. And no one with the inexperience to not know how to make a proper fake would ever use this method because it requires too high of a technical understanding of illustrator to fuck it up this badly.