Don't be fooled by the 36 comparisons. They are not independent comparisons. Once you know that you match person X on this little area of chromosome 9, then it immediately becomes *much* more likely that their child has that same segment. And in that case, you will match the child, and also the parent will match the child. For the less close relationships, the "*much* more likely" becomes less likely. But still, there are a lot of 5cM segments in the entire genome. If you look at just the group of close family members to X, there are probably quite a few segments larger than 5cM that they all share by being close relatives. So all of their comparisons will show matches on such segments. You just needed one little area of your genome to happen to match one of those segments they all share, and that may be what happened on chromosome 9.
I'd say you have good evidence that X's 5cM segment is real -- i.e., it doesn't jump between her maternal and paternal copies of chromosome 9, but is a real, contiguous segment on one chromosome that she got from one parent. It would still be possible that your own segment which matches actually jumps between your maternal and paternal chromosomes.
The fact that your sister matches on the same segment reduces the possibility of the latter. But on about 1/4 of your genome, you and your sister are "fully identical," meaning you have exactly the same maternal and paternal chromosomes. Your mother gave you and your sister the segment from the same one of your maternal grandparents, and your father gave you both the segment from the same one of your paternal grandparents. So there is still a decent chance your are fully identical on this part of chromosome 9, in which case both of your matches with X and her close relatives could be pulling from both maternal and paternal chromosomes.
There is a simple way to check if you are fully identical. You and your sister could either upload to GEDmatch or else upload both kits to David Pike's tool "Search for Shared DNA Segments in Two Raw Data Files"
Their comparisons where show you "Fully Identical Regions" (FIRs). If you are half-identical in that part of chromosome 9, which is about a 50% chance, then that would be evidence that the segment is real.
But at that point, it could still be possible that the segment was very common in your ancestral population, in which case there might be other ancestors who could have sent it down to you, not just the one you have identified. But if the ancestral population has only a small amount of endogamy, then that's not much of a worry.
(BTW, I highly recommend identifying where you are half-identical and fully-identical with your sister. That information can be very useful for mapping the different parts of your genome to your four grandparents, which then makes it easier to cousin-match -- you already know which quarter of the tree to be looking in.)