So, what has been discredited, that it was published in 1595 or that it correlates with 112 successive Popes?
I don't have time to look up the links now but the early papal prophecies in the series are thought to be authentic, written either by St. Malachy or sincerely on his behalf. But the later prophecies, written in a very different style, are forgeries added along the way, many about popes who were already deceased.
Of course, if the 'prophecy' was actually composed in the 16th century, it makes sense that all the descriptions of the popes between the 1100s and 1590 would be accurate. And those descriptions are not only more accurate, they all focus on such verifiable matters as the pope's coat of arms, family, or birthplace.
The second problem is that the later "descriptions" (unlike the ones before 1590) are so short (just 2-3 words) and vague that you can read pretty much anything into them that you want. Most of them are a real, serious stretch, and a couple are so cryptic that interpreters don't even try and leave them blank.
The third problem is that even the manuscript doesn't claim to list ALL the popes - just a bunch of them, and then the last one, "Peter the Roman". Which corresponds nicely to the first pope . . . Peter.
As Cecil Adams commented about Nostradamus, I think this one does for B.S. what Stonehenge did for rocks.