You're assuming that the word "vertex" has some sort of an unambiguous higher meaning, when it really doesn't.

When you take a word such as vertex, which has a very clear definition for polygons, and try to apply it to something more general you'll usually find that it has multiple facets which are simultaneously true for polygons (eg. vertices are non-smooth points, (~ sharp turns) and between any two vertices is a straight line), but cannot be simultaneously generalized to circles.

If you look for non-smooth points on a circle, you'll find none, but if you look for points separating straight lines, you'll either find that that doesn't make any sense, or that ∞ might be a sensible answer intuitively (as you do when you look at a circle as a limit of polygons with increasing number of sides).