TL;DR: Lucas' original idea for the Clone Wars was a series of skirmishes and battles lasting several years, with the clones acting as antagonists rather than the Republic's army.
Out-of-universe, it is a continuity error. There was originally supposed to be a much longer period between the original trilogy and the events that preceded the movies. In that version of events, "the Clone Wars" referred to a series of skirmishes and uprisings that occurred over several years in rapid succession. Bail Organa, along with Garm Bel Iblis, were Republic leaders whom the Jedi served under, and Anakin's slide to the Dark Side was apparently much more gradual than what we witnessed. Unfortunately, most of that story was fleshed out in EU material and was never made canon or official by any source.
Also, the original idea for the Clone Wars - according to notes that Lucas had given author Timothy Zahn - was that the clones were the antagonists of those conflicts. Zahn used this concept in his original Thrawn trilogy, with Imperial captain Pellaeon reflecting back on those dark times:
No matter how many times he saw it happen, he would never get used to
these sudden dips into the slippery twilight of clone madness. It had,
he knew, been a universal problem with the early cloning experiments:
a permanent mental and emotional instability, inversely scaled to the
length of the duplicate's growth cycle. Few of the scientific papers
on the subject had survived the Clone Wars era, but Pellaeon had come
across one that had suggested that no clone grown to maturity in less
than a year would be stable enough to survive outside of a totally
Given the destruction they'd unleashed on the galaxy, Pellaeon had
always assumed that the clonemasters had eventually found at least a
partial solution to the problem. Whether they had recognized the
underlying cause of the madness was another question entirely.
Needless to say, Lucas changed all that when writing the prequels. Whether by mistake or design, Lucas severely streamlined events, instead asking us to believe that the Jedi Order went from thousands to a handful within a 3-year period. Or that, in the same period, Anakin went from a whiny, brooding teenager to a Sith Lord capable of hunting down and defeating Jedi Masters in single combat.
In-universe, the only explanation I can come up with is that it's a grammatical thing. In a similar example, I point out the "Stark Hyperspace War", which was really only a single battle lasting a few days. Perhaps a long, long time ago they had a penchant for exaggerating such things.