Only man (male and female) has an immortal soul, and only man is made in the image and likeness of God.
When Jesus Christ, the Second Person of the Holy Trinity, became man, thus divinity and humanity were united, and men and women were redeemed and saved. When a human dies, only their body dies; their immortal soul continues to exist. Particular judgment happens immediately and the soul goes to heaven, purgatory, or hell.
We do know that only angels and human beings can go to heaven, since only angels and man have immortal souls. Animals have souls, but not immortal ones. Animals were not made in the image and likeness of God, and Jesus Christ is not their Lord and Savior. Does this mean that those in heaven will never see Sylvester the cat or Spot the dog ever again? No; the Church has never taught nor decided what happens at the end of time.
While not by dogma, doctrine, or even official teaching, most theologians speculate that after resurrection of all the dead, the just and righteous souls in heaven will have their glorified bodies reunited with their immortal souls, and God will also allow us to enjoy our former pets as we will also enjoy good food, music, and all that is good that God created.
Only the damned in hell, who will also have their bodies reunited to their immortal souls, will be deprived of any pleasure or happiness. They will get bad food, smell rotten odors, hear horrible noise, and feel painful cold and heat. Their beloved pets will not join them (but the nasty neighbor’s dog just might).