Well, regardless of whether their stuttering is the consequence of an actual natural speech impediment or is caused by Rick's behavior as an alcoholic and Morty's severe anxiety and insecurity (as /u/schwarze_banana points out), they both do have some obvious stutter. Relating to your question: both characters are voiced by the same voice actor. I remember reading an article explaining that the unfluencies in his voicing work (as well as Rick's constant burping) are the consequence of his ritual of drinking a few beers prior to recording. I'd say this is a clear indication of the fabricated nature the characters' stuttering behavior, so I personally am not too affected by it (like OP appears to be). But nevertheless it's great to read that OP appreciates it to this high extent! Besides that, it is an interesting topic to discuss on this sub, if you ask me.