Well, your 23 g-grandfather might very well be an ancestor you share with your match, as well as with literally millions of other people. I can pretty much guarantee it is not your most recent common ancestor (MRCA) and the DNA test is not matching you because of that. You have over 8 million 23g grandparents, so there is no way to trace something in you to that particular person.
I may be wrong on this so people can correct me, but the DNA matching and the family tree matching are two separate things. Ancestry takes your DNA and compares it with the DNA of others, and when enough things they look at match they consider it a hit. When you click on that hit, they then compare your two trees, but that has nothing to do with what they found in the DNA. They know who you are in your tree, and they know who your match is in their tree, and they just simply walk up the tree looking for a common ancestor or ancestors by name. When you get a hit and you can't find a common ancestor, that just means neither of your trees has identified that person yet. It can also mean the match is incorrect. Think of the probabilities ancestry shows like the weather forecast. If they say 50% chance of rain, it may rain and it may not. The same goes for a 50% probability of a DNA match - it may be a match, and it may not.