A sphere can be seen as a plane plus the point at infinity. So, what is a sphere with a hole? This is not topologically equivalent to a sphere, but this is equivalent to a plane (the sphere is the plane plus the point at infinity, you remove it, you get the plane back).
Now you add a new hole, this is your plane with a hole, which is also a sphere with two holes. However, on the sphere with two holes I can draw two unshrinkable loops (around each hole), on the plane with a hole, there is an unshrinkable loop around the hole, but what about the other one?
The two loops around each hole are homotopic to each other, so you really have only one loop, just like in the plane case.
Think of the two holes as being at the north and south poles of the sphere. There is one unshrinkable loop: the equator. A loop around either pole can be moved to match the equator.