Moriarty uses no other name in human or Uratha society. He likes the sound of it, as well as the air of menace he feels it implies. Moriarty sees himself as a dark hero out of the movies and his pack as a band of outcast misfits who get results.
Solo has nearly been a member of every pack in Seattle. Her perchant for individualism has not gone down well with Alpha's in the past and has led to her being booted out of numerous packs. This has not stopped her staying true to herself.