Over the course of Warner Brothers animation history, there have been some truly amazing parodies and crossover with the heroes of the DC universe. This was not one of those. Instead, the use of the word "robins" during a dramatization of Shakespeare's Midsummer Nights Dream led to a poor translation of Batman and Robin saving the day for all of 3 seconds. It appears that the Robin in question was Tim Drake - the sidekick in the comics in 1993, though strangely it was Dick Grayson who was animations Robin of the day in Batman: The Animated Series.
But wait, there's more.
In the earlier episode, Taming of the Screwy, the Animaniac party crashes scored intros with the Dark Knight himself as well as the felonious feline Catwoman.