Creating specific objects based on user input

You may go for polymorphism with factory pattern:

public abstract class Pet {
  private String name;
  private int age;

  protected Pet(String name, int age) {
    setName(name);
    setAge(age);
  }

  public void setName(String name) {
    this.name = name;
  }

  public String getName() {
    return name;
  }

  public void setAge(int age) {
    this.age = age;
  }

  public int getAge() {
    return age;
  }
}

public class Dog extends Pet {
  public Dog(String name, int age, param3, param4) {
    super(name, age);
  }
}

public class Cat extends Pet {
  public Cat(String name, int age, param3, param4) {
    super(name, age);
  }
}

public class Fish extends Pet {
  public Fish(String name, int age, param3, param4) {
    super(name, age);
  }
}

public class PetFactory {
  public Pet newPet(String petType, String name, int age, param3, param4) throws IllegalArgumentException {
    if ("Dog".equalsIgnoreCase(petType))
      return new Dog(name, age, param3, param4);
    if ("Cat".equalsIgnoreCase(petType))
      return new Cat(name, age, param3, param4);
    if ("Fish".equalsIgnoreCase(petType))
      return new Fish(name, age, param3, param4);
    throw new IllegalArgumentException("Unknown pet: " + petType);
  }
}

If you have a frame UI, you can create a radiobutton group to choose the type of pet and text boxes for pet parameters (of course, you need to check user input and give feedback). It would be better to provide your data restrictions on labels. If you use console, for pet choosing provide user an opportunity to choose between "cat", "dog", "fish", etc. Also provide your restrictions about other parameters. Control his input and give feedback. There plenty ways of realizing it, specify your vision.

Take a look at decorator pattern.

abstract class Pet {}
abstract class PetDecorator extends Pet {
    void initialize() {
        // read name and age from console
    }
}
class CatDecorator extends PetDecorator {
    @Override
    void initialize() {
        super.initialize();
        // read cat specific fields
    }
}
// other classes omited for brevity
// inside Main
List<Pet> readInfoAboutPets(int numberOfPets) {
    List<Pet> pets = new ArrayList<>(numberOfPets);
    for (int i = 0; i < numberOfPets; i++) {
        // read pet type from console
        switch (petType) {
            case "cat":
                PetDecorator petDecorator = new CatDecorator();
                petDecorator.initialize();
                pets.add(petDecorator);
                break;
            // case rest
        }
    }
    return pets;
}