Is this a fallacy, and if so what kind of fallacy?

I see two problems here. First, the listener is presenting a argument. The person mentioning the Bible is referencing a story told by a known author. The claim that the author is really known could be up for debate, but the listener isn't debating that. Instead, the listener is responding to a claim not made... that the story they are referring to is true even though the author is unknown.

Next, we have a Non Sequitur . The listener's conclusion, that "the story is not true" does not follow—assuming that the story was told by an anonymous author. The reason given, "the real authors of the different books of the bible are not known" assuming true, would not mean the story is false. The assumption here is that anonymous authors can only tell false stories, which is clearly fallacious. Anonymity of authorship does cast doubt on the alleged facts presented in the story, but in no way can one conclude that the story must be false.