Sarah, when you said he listed his birth on all those things he completed in the U.S., what are those things? Basically, I'm asking for the source documentation. Because it's kinda weird to write November 1884 on American documents. They want the full date. So what are these things he completed?
If your grandmother gave her father's birth for a local history book, I'd file that one under circumstantial evidence. Or just iffy. Your grandmother wasn't there when he was born. It's a clue, but I'd keep looking for other clues.
Yes, you'd have to read all the 1880s for every single Manuel born. That's the needle in the haystack approach. No one on the list has him in their database, otherwise they would have said so.
DNA has the potential to point you to a freguesia, but it depends on how many testers from that freguesia have tested thus far.