a square has infinite points too so that explanation is pretty poor, and besides it doesn't even help define any "sides"
pretty much the only real answer is that a circle is infact not a polygon, and thus has undefined amount of sides
i mean if you were to like approximate it really hard, a circle CAN be defined as the limit of n-sided regular polygons and thus has infinitesimal sides but i dont really know enough topology to argue this
using the power of google i have concluded that the latter is not a proper explanation as that is actually an apeirogon, not a circle.