Well, first of all, yeah, I think it would be more credible if he meant the Bible in general, not just the AV/KJV.
But secondly: remember that up until the perfecting of movable-type printing, there was no true mass production of books. And one guess as to what was the first book Mr. Gutemberg got off his press.
Then, for quite a while after that, indeed for most of the world until the last century, widespread literacy remained the exception rather than the norm.
So, for certain, the Greek Classics and the Bible (first as Jewish Scripture, then as that AND as the OT/NT combo) were about at the same degree of accessibility to the masses all through the last 2 thousand years, and the Bible has certainly been more published over the last 500.
The Chinese and Greek classics were indispensable texts for the educated Chinese and European… who was a member of a tiny Learned Elite. OTOH, for the last 400 years your average schnook can’t stick his head out of the ground anywhere w/o some preacher/missionary/televangelist/cult leader shoving a Bible in his face; it at times being sole textbook for the only school of any kind within two weeks’ overland travel. Add that at least parts of the Bible have been translated into virtually every language (I doubt there is an edition of Aristotle’s Politics in !Kung) and you can see it getting close.
As for the Chinese classics, at any given time 1/4 to 1/5 of humanity has lived in “China”… but not all of them had access and they were not read anywhere beyond that immediate region – while the Western canon has been spread worldwide over the last 300 years, usually with missionaries and their Bibles close behind or even leading the way.
So it does not take a major effort to conceive of the possibility of the Bible being the most widely circulated book on Earth.
Which brings up an interesting element… the Bible may be the “most purchased” book, but not necessarily purchased by the individuals – rather, by the churches, who then give it away, and that may in turn make it the most-owned book, but not necessarily the most regularly read, as a lot of the faithful only crack it open at church, if ever.