We may start with this definition of the circle:
The set of all points in the plane that are at equal distance (different from $0$) from some fixed point in that plane is called a circle.
It is self-evident that only curve that does the job is the circle.
How to prove that only the circle does the job?
Added: On second thought, I am not sure that there are no curves that do not have tangent anywhere that would also qualify to satisfy the definition of the circle. Now it is not self-evident any more.
If this is your definition of a circle, then a circle is whatever it specifies. If we give the plane the Manhattan metric, where the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $|x_1-x_2|+|y_1-y_2|$ circles look like squares that are rotated $45^\circ$ from the axes. If we give the plane the discrete metric, where the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $1$ unless the points are identical, the circle of radius $1$ is the whole plane except the center and the circle of any other radius is empty.
What I think you are trying to ask is to prove your stated definition with the usual metric matches some intuitive definition of circle that you have. If so, we need to understand your intuitive definition to make progress. Your stated definition is my intuitive definition, so they match easily.