Every born-again person, every Christian, has an identity in Jesus Christ, they are "new creatures" in him (2 Corinthians 5:17). And it's to Jesus, not to some animal, that God wants you to be conformed.
Romans 8:29
29 For those whom He [God] foreknew, He also predestined to become conformed to the image of His Son...
2 Corinthians 4:7-11
7 But we have this treasure [the Spirit of Christ] in earthen vessels, so that the surpassing greatness of the power will be of God and not from ourselves;
8 we are afflicted in every way, but not crushed; perplexed, but not despairing;
9 persecuted, but not forsaken; struck down, but not destroyed;
10 always carrying about in the body the dying of Jesus, so that the life of Jesus also may be manifested in our body.
11 For we who live are constantly being delivered over to death for Jesus' sake, so that the life of Jesus also may be manifested in our mortal flesh.
How is your furry "identity" helping you live out your true identity in Jesus? How is identifying with what is lower (an animal) help you to live in the highest of ways, as a child of God Almighty?
God is preparing all of His children for an eternity to come with Himself. It is this forever existence beyond our earthly one that is the Real Life for which God has made all human beings. That life will be centered upon Him, not cats, or dogs, or family, or sports, or whatever. God is making us ready for the unending Real Life to come by teaching us now to be focused upon Him, not upon ourselves, or the noisy, corrupt World around us, or upon the dark deceptions of the devil. The more we are crowded up to God, desiring Him above all, the better prepared we are for the eternity to come. Does your furry preoccupation help make you more like Jesus? Does it prepare you for the eternity to come in which being a "furry" will have no place?