Rabbi Moshe Shapiro was in error. We now have scientific evidence that other primates laugh in the same way that humans do. However, Rabbi Shapiro should not be held accountable for his error since this evidence was only recently confirmed and has yet to penetrate the general knowledge base outside the scientific community. And as right now, only a few species of primates have been confirmed to laugh like humans do. Other animals often make a laughing sound but we haven't been able to prove that this sound functions the same way as our laughter, despite sounding similar.
Source: BBC Via Youtube https://www.youtube.com/watch?v=hhlHx5ivGGk
Note: We can only characterize laughter as either a sound, or a function. If it's a sound, then lot's of animals laugh. If it's a function, then many animals can be said to laugh with other modes of body language. If it's a mix of a specific sound related to human laughter, and the function of human laughter, then only certain primates can be said to laugh.