Yes, clones have fingerprints. No, they wouldn't be exactly like their original: fingerprints are influenced by in-utero and epigenetic factors, not just by the DNA. A clone's fingerprints would be very much alike its progenitor (and some features would be nearly identical), but not the same.
Copies, i.e. identical living beings created by some kind of duplicating machine, (faulty) transporters, and time loops, would have of course the same fingerprints - unless the machine for some reason is designed to do otherwise. Even though the two beings would not be identical since they're exposed to different stimuli and environments (so that they would immediately start diverging mentally and biochemically), and in the end it would be possible to tell them apart, still having only one of them at hand one could not say whether it's the original or the "copy" (indeed, even the concept of "copy" gets a bit murky).
If the two beings started with the same set of fingerprints, barring accidents, wounds and similar more or less intentional alterations, they would go on having the same fingerprints; that alone could not be used to tell them apart. Accurate chemical analysis of said fingerprints possibly would.
Telling identical fingerprints apart
Several chemical and hormonal contaminants exist that might end up in fingerprint grease in detectable amounts (given sufficiently advanced technology).
Also, the isotopic composition of said grease might yield valuable clues. For example living long enough in different solar systems would alter the relative isotope abundance in finger grease (which is mostly CHON), and that in turn could allow to tell the fingerprints apart.
Different Population I stars might sport different ratios of some elements (e.g. 14N and 15N). Living, breathing and eating proteins on such a star's planet would slowly alter an organism's isotope proportion to reflect that of the host planet.
Also, a young star's planet should experience less cosmic ray activity, therefore the abundance of 14C in the atmosphere would be lower. Planet climate, as well as the latitude at which the clone lived, also affects relative abundance of 18O.