Can multiple Digital Signature fields in PDF form be auto-filled?

The digital signature always applies to the whole document, and any marks it makes on a page are irrelevant. Never, ever, should there be more than one digitial signature on the same file by the same person. So the question might be: how can I initial all pages of a file BEFORE applying a digital signature... 

What do you mean by "form filling app software" and what does that have anything to do with applying a digital signature ?

You  sign one signature field and you can initial all pages in a document in a variety of ways using Acrobat JavaScript.

All you need to do is add a script that executes when the signature is applied to grab the initials from the first, middle, and last mame, and copy to the text fields that you need them to be copied to.

This is an entire different procedure than trying to copy one digital signature into different signing fields throughout a PDF.

If you're using software that copies a digital signature and populates it in other fields of a PDF document, I may have to say that you're not protecting or enforcing digital signature standards appropriately.