You can use merge multiple stream and drop duplicates.
For the first dog of a given name, you can do
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import java.util.stream.Collectors;
import java.util.stream.Stream;
public class A {
    public static void main(String[] args) {
        List<Dog> dogList1 = Arrays.asList(new Dog("a", 1), new Dog("b", 2), new Dog("f", 3));
        List<Dog> dogList2 = Arrays.asList(new Dog("b", 4), new Dog("c", 5), new Dog("f", 6));
        List<Dog> dogList3 = Arrays.asList(new Dog("b", 7), new Dog("d", 8), new Dog("e", 9));
        List<Dog> dogs = new ArrayList<>(
                Stream.of(dogList1, dogList2, dogList3)
                        .flatMap(List::stream)
                        .collect(Collectors.toMap(Dog::getName,
                                d -> d,
                                (Dog x, Dog y) -> x == null ? y : x))
                        .values());
        dogs.forEach(System.out::println);
    }
}
class Dog {
    String name;
    int id;
    public Dog(String name, int id) {
        this.name = name;
        this.id = id;
    }
    public String getName() {
        return name;
    }
    @Override
    public String toString() {
        return "Dog{" +
                "name='" + name + '\'' +
                ", id=" + id +
                '}';
    }
}
prints
Dog{name='a', id=1}
Dog{name='b', id=2}
Dog{name='c', id=5}
Dog{name='d', id=8}
Dog{name='e', id=9}
Dog{name='f', id=3}
In each case, you can see the first instance of a name is retained.
For unique names
Set<String> names = Stream.of(dogList1, dogList2, dogList3)
                          .flatMap(List::stream)
                          .map(Dog::getName)
                          .collect(Collectors.toSet());