The material in fridge magnets is magnetised in strips that run the length of the material, forming a Halbach Array. The result of this is that the magnet has a reasonably strong magnetic field, but this field extends no more than a millimetere or two beyond its surface. That's why you cannot use a fridge magnet to affix more than a few pages of paper to your fridge.
To see this for yourself, stick two fridge magnets together, back to back. They will slide against each other in one direction. If you slide them at right angles to that, they will alternately attract and repel one another.