How many right circular cylinders can pass through 5 general points in ℝ3 ? Edit: 0,2,4, or 6.
If the 5 points are generic then I believe Lemma 7.1 in arXiv: 1306.2346 is relevant to your question.
The Lemma, for the case of cylinders, says if there are two cylinders that pass through the same 5 generic points and there is an isometry that maps one of the cylinders onto the other, then the two cylinders are the same.
(This is only an answer rather than a comment because I am not allowed to post comments.)
Edit: there's also a relevant paper called "On circular cylinders by 4 or 5 points in space" by Devillers, Mourrain, Preparata and Trebuchet.