public qweqwe() {
    this.map = new HashMap<Person, List<Pet>>();
}

Person [name=Yurii]
       Pet [nickname=Margo, vyd=Dog]
       Pet [nickname=Chita, vyd=Dog]
Person [name=qweqwe]
       Pet [nickname=Chita, vyd=Dog]

void remove(Map<Person, List<Pet>> map, String name, String nickname) {
    Maps.filterKeys(map, person -> person.getName().equals(name)).forEach((person, pets) -> {
            pets.removeIf(pet -> pet.getName().equals(nickname));
    });
}

void remove(Map<Person, List<Pet>> map, String name, String nickname) {
    map.entrySet().stream().filter(person -> person.getName().equals(name)).collect(Collectors.toMap(p -> p.getKey(), p -> p.getValue())).forEach((person, pets) -> {
                pets.removeIf(pet -> pet.getName().equals(nickname));
    });
}

public class Person {
    private String name;

    public Person(String name) {
      this.name = name;
    }

    @Override
    public boolean equals(Object o) {
      if (this == o) return true;
      if (!(o instanceof Person)) return false;
      Person person = (Person) o;
      return name != null ? name.equals(person.name) : person.name == null;
    }

    @Override
    public int hashCode() {
      return name != null ? name.hashCode() : 0;
    }
  }

public class Pet {
    private String nickname;
    private String vyd;

    Pet(String nickname, String vyd) {
      this.nickname = nickname;
      this.vyd = vyd;
    }

    @Override
    public boolean equals(Object o) {
      if (this == o) return true;
      if (!(o instanceof Pet)) return false;
      Pet pet = (Pet) o;
      if (nickname != null ? !nickname.equals(pet.nickname) : pet.nickname != null) return false;
      return vyd != null ? vyd.equals(pet.vyd) : pet.vyd == null;
    }
  }

ListMultimap<Person, Pet> map = ArrayListMultimap.create();

Person personFirst = new Person("name1");
map.put(personFirst, new Pet("nick1", "vid1"));
map.put(personFirst, new Pet("nick2", "vid2"));
map.put(personFirst, new Pet("nick3", "vid3"));

Person personSecond = new Person("name2");
map.put(personSecond, new Pet("nick1", "vid1"));
map.put(personSecond, new Pet("nick2", "vid2"));
map.put(personSecond, new Pet("nick3", "vid3"));

map.remove(personSecond, new Pet("nick3", "vid3"));

Map<Person, Collection<Pet>> mapResult = map.asMap();

public boolean put(@Nullable K key, @Nullable V value) {
   return this.get(key).add(value);
}

public boolean remove(@Nullable Object key, @Nullable Object value) {
   Collection collection = (Collection)this.asMap().get(key);
   return collection != null && collection.remove(value);
}