For anyone with a suffix at the end of your name when applying for the AppleCard, leave it off
So I was trying to apply for the AppleCard today and I have the III at the end of my legal name, it is on my license too. I kept getting the error when verifying my identification. Called Apple, they got me in contact with Goldman and they had no idea. In a last attempt to try, I left the III off of my application, and I was approved immediately.
After this I texted Apple to try and amend my name after being approved and they say that they do not currently support adding a suffix.
So if you are having the same issue, try this.