وَلَدُ النَّاقَةِ initially suggested to the listener a very young animal unsuitable for riding. The Prophet ﷺ reminded him that walad denotes offspring, whether young or grown.
The humor did not require lying or humiliating anyone. If someone misunderstands a lighthearted remark, explaining it kindly is part of good character.
For example, in class, a teacher may use gentle wordplay to clarify a term without making the learner the object of ridicule.