AI therapist is something of a poetic misnomer. Of course, AIs (as distinct from AGIs) do not develop emotional problems. Simple AIs break hard, either realizing there is a problem and putting themselves on standby, or not, and carrying out meaningless tasks. More sophisticated AIs break in more subtle ways, often not failing completely but carrying on actions that in the wrong contexts, or being unable to access any functions other than fuzzy logic, with unpredictable results.
AIs based actively learning neural networks or evolutionary algorithms often develop a few strange habits based on coincidences or improperly optimized reward systems. AI therapists can expect to treat surveillance systems who have decided to mark every person wearing blue as a unique morph, automechs that just won't stop insulating, servitors that obsessively imitate the actions of their owners, and the occasional neurotic elevator.