“Have this mind among yourselves, which is yours in Christ Jesus, who, though he was in the form of God, did not count equality with God a thing to be grasped, but emptied himself, by taking the form of a servant, being born in the likeness of men. And being found in human form, he humbled himself by becoming obedient to the point of death, even death on a cross.” 
– Philippians 2:6-8

It is not irrational to compare the condescension of our Lord Jesus to a man becoming an animal. Still, it is only a comparison, and in this case, the measure might be greatly lacking. We could say that God becoming man is like man becoming an ape. Just as an ape lacks the ability to be self-aware, to communicate verbally, or to manufacture machines, so mankind could never relate to the divine.
For that reason, Jesus became human to reveal God to us creatures. A man becoming an animal could easily relate to other animals of the same nature. The curious thing is, what person would voluntarily lower himself to become some “natural brute beast?” Such a thing happened once: Nebuchadnezzar, because of his great pride, was given the nature of a beast and lived among them and ate grass like an ox. When his wits were restored, he humbled himself before God. (Daniel 4:28-37)
Considering the great self-humbling of Jesus, who humbled himself and became obedient unto death, even death on a cross, should produce deep humility within us. And if you own a pet, the next time you have dealings with it, breathe a prayer of gratitude for the great step downward Jesus took for you.