Actually, yeah, it is possible; magnets do affect their surroundings, even the non-metallic ones...
The theory goes, if you leave a piece of iron in a really strong magnetic field for a while, it will become pretty well magnetized. It has to do with the magnetic 'domains' (I think they're called); basically, a piece of metal is divided into regions where the magnetic field points in one direction. Most of the time, this is random, but if it is subjected to another magnetic field (or certain forces - such as hammering a nail in the northerly direction) then they will align with each other and the metal will become magnetic. If it is heated, the atoms of the metal become loose and spinny and the domains are re-randomized.
Now, if you had a particular magnetic pattern passing through a slot several times per day, over the course of a year it would become pretty well-magnetized, and unless the pattern is changed fairly often, it would stay that way. This device simply picks up the residual magnetism and attempts to reconstruct the original pattern.
As far as fingerprints go, it all depends on how fancy you want to make them. We've had the technology to recognize fingerprints in a handheld device for about ten years now, as far as I know, and one could assume that in a sufficiently advanced technological setting, miniaturization to the point of fitting on a swipe card could be achieved. Powering it could potentially be a problem, but you could just say that the body heat from having the hand on it is enough to power a hardwired circuit that would confirm recognition.
Another option would be to devote a certain section of the magnetic pattern to a digitalization of the finger print reading, such that only the correct fingerprint would generate the appropriate magnetic field.