No, that's nonsense. If a clone had a soul, in the narrowest sense, we could have a bunch of people running around with identical souls. Which raises the question, what exactly is a soul anyway?
The word "soul" didn't even exist in the times of Jesus, Socrates or Aristotle. The Biblical Hebrew word for soul is 'nephesh' meaning life, or vital breath. So soul was just a synonym for aliveness, breathing.
Not that the concept of some spirt-like soul isn't widely held.
Muslims, from the hadith, believe Allah assigns an angel to "breath" soul into an embryo after 40 days of pregnancy. Hindus believe the soul, jiva, is the immortal essence of a human which reincarnates again and again until it becomes perfect. In Ayyavazhi, the soul is believed to be in a flame shape and its symbol is smeared in powder between the eyebrows as a ritual called Thirunamam. The ancient Chinese believed every person's soul was composed of seven p'o (visible personalities attached to the body), and three hun (ethereal complements not necessarily tied to the body—the hun wanders, and if it stays away death results).
Whatever a soul is, as Arthur Caplan, chairman of the Department of Medical Ethics at the University of Pennsylvania delicately puts it, "If humans have souls, then clones will have them, too."