The name’s the same does not mean the ancestor is the same, even if descendants of those “same name” people share DNA.
The way AncestryDNA‘s shared “hints” work is that if you share DNA, have posted your online tree, and you and the person with whom you share DNA has posted their tree and your trees “share a name without dates,” that person is your “shared hint.”
The problem is that many people assume they are more than hints.
They are not.
Sometimes are not even good hints at that if one looks at the supposed children of the hint–sometimes they are born nearly 100 years apart. Difficult for a man to do (even if he’s really precocious and remains in excellent health his entire life) and impossible for a woman. And the hints, by their very nature, cannot look at the parts of your trees that are incomplete.
As we’ve mentioned before these hints are suggestions. Even if they are correct, you and that person may share additional people in your pedigree and the “hint” may not even be the family through whom the actual DNA is shared.
Remember that the “hints” are “hints,” not fact.