It's easy to figure out why triangles have unique circumcircles; take two points on a side and look at the family of circles passing through them, only one of which (and one of which always) passes through the other point;
I can't produce a similar reasoning for incircles, however. Can someone help me figure it out?
Whichever way you take two sides of a triangle, the centres of the family of circles that are tangent to both lie on their angle bisector;
And you can always adjust the size of the circle (by taking different centres along the bisector, each of which belongs to a circle with a unique size) to get the evidently singular one that's also tangent to the third side.