It’s a nice, well-worn fantasy. Hope to the hopeless, comfort to the desperate, all of that. Especially suggestions that the animals will change their nature and complete peace and harmony will abound as we live forever. The utopian ideal. Poetic.
Sadly, it is but a wishful product of human imagination. Forever is a term for which the real meaning eludes most people’s grasp.
I disagree that the world would become overpopulated. If there were no death ... odds are that birth rates would be extremely low. We see that happen when infant mortality is low, and in areas where there is easy access to healthcare and medicines.