Yes, they rhyme.
The name of the rhyme that starts with "brown bear brown bear what do you see?" is exactly the same as the phrase in quotations. There is no other name for this rhyme.
Hare, Brown Bear, Polar Bear, Grizzly Bear, Panda Bear.
Bear Hare
bear
Claire Cher
Marilue. has written: 'Bobby Bear's Christmas' -- subject(s): Bears, Christmas, Fiction, Stories in rhyme 'Bobby Bear and the kite contest' -- subject(s): Animals, Bears, Fiction, Kites, Stories in rhyme 'Bobby Bear's Thanksgiving' -- subject(s): Animals, Fiction, Stories in rhyme, Thanksgiving Day 'Bobby Bear and the Friendly Ghost' 'Bobby bear at the fair' -- subject(s): Animals, Bears, Fairs, Fiction, Stories in rhyme 'Bobby Bear's red raft' -- subject(s): Stories in rhyme
care bear
bear's cares
bear's cares
bear's cares
deer,dear bear,bare