I don't think he got it wrong, since the kanji is actually used for counting. But in both instances that I can see, a number is very clearly shown.
Again, the example of "That mountain is five times as high as this one.", the original Kanji being "あのやま山はこのやま山の5ばい倍のたか高さである。". You'll notice the bolded part is our kanji in question, with a 5 right besides. The other example, when it is used as in "1 in n" has the original kanji as "40倍", once more with the number right besides the kanji we are discussing, this one meaning "1 in 40".
"速度倍化イルバニーア" This is the original kanji of Ile Vernier, the speed one. You'll, again, notice there's no number. This seems about the same with "かれ彼はわたし私のばい倍た食べた", which translates to "He ate twice as much as I did.", which is meant to show the meaning of the Kanji when used as a noun.
Therefore, from the comment of Lehen and comparisons, it seems pretty clear to me that the kanji is being used as a noun, which is when it is meant to signify "double/two-fold".
Though anyone that knows japanese can say if I am wrong in my assumption.