Are similar circles really a thing?

You're right: any two circles are similar (and so there's not much point of talking about "similar circles")! In general, two shapes are "congruent" if you can turn one into the other by translations (moving around in the plane), rotations, and reflections. Two shapes are "similar" if you can rescale ("zoom in or out" on the picture) one of them to turn it into a shape that is congruent to the other. Given two circles, you can rescale one so that it has the same radius as the other, and then any two circles with the same radius are congruent since you can just translate the center of one to the center of the other.

Yes indeed. Every circle is similar. You can always scale one of them to match the other. Actually, this is the definition of similarity. In case of triangles, this definition yields the result that the sides are proportional. "The sides of one triangle are proportional to the other" is not the actual definition of similarity. You may have a look here