Bugs plays tribute to the piano entertainer Liberace before starting to play Chopin's Minute Waltz. Note the candleabra on the piano, a Liberace trademark. All that's missing are the sequins...
The title is a word play on the expression "hide nor hair", as in "I haven't seen hide nor hair of him", meaning "I haven't seen any trace of him". "Hyde" in the title refers to Mr. Hyde, the evil persona of Dr. Jekyll in "The Strange Case of Dr. Jekyll and Mr. Hyde".
Both Bugs and his benefactor break the fourth wall early on.
"Mister Hyde" has more than a passing resemblance to the animated version of the Grinch, which was overseen by Warner Animation great Chuck Jones.
The meek little gent might be an Americanized version of Peter Lorre, though he also resembles future Laugh In great Henry Gibson.