Yup, Umbran is correct.
I'm an MA in clinical psychology working toward my doctorate (just a dissertation and internship to go).
From this webpage (but also from the DSM...I don't know if I can copy the DSM, but quoting a web page seems fine.)
Diagnostic Criteria for 312.32 Kleptomania | Lori Riddle-Walker MFT, Treating OCD and Related Disorders, San Diego
Diagnostic criteria for 312.32 Kleptomania
A. Recurrent failure to resist impulses to steal objects that are not needed for personal use or for their monetary value.
B. Increasing sense of tension immediately before committing the theft.
C. Pleasure, gratification, or relief at the time of committing the theft.
D. The stealing is not committed to express anger or vengeance and is not in response to a delusion or hallucination.
E. The stealing is not better accounted for by Conduct Disorder, a Manic Episode, or Antisocial Personality Disorder.
Sociopathy is actually an extreme form of Antisocial Personality disorder, hence it disqualifies someone from having kleptomaina.
An important note about Kleptomania is that the stealing is ego-dystonic (which means that the behavior is unpleasant and not enjoyable for the person doing it). They don't want to steal, but can't help it.
Though, Kleptomania is about the act of taking, specifically...not about saving. Compulsive hoarding (saving/being unable to get rid of things) is another consideration, which is likely to be a discrete diagnosis in DSM-V. One could have both kleptomania and compulsive hoarding, which would be a very unfortunate pair of comorbid diagnoses.