My niece just got a 4 mth old maltese Shi Tzu puppy and she is very clingy.
When my niece is around the puppy will not go to anyone else and shows some aggression when someone else tries to take her.
My niece loves the fact that the puppy loves her so much but wants the other family members to be important to the puppy also.
Absolutely! It's understandable that your niece should feel flattered that her puppy loves her so much, but that's really a human interpretation of the situation.
This 'clingy-ness' isn't really love, more an attempt at domination (or the pup expressing her authority) in my opinion.
If the puppy were to be very closely attached to your niece in a way that made her very upset, and whine and cry when apart from her. That would suggest a separation anxiety problem.
If she were to cower, shake, growl and snap as if cornered or afraid when approached, that would indicate poor nerves and a potential 'fear biter'.
However, what you're describing in the way that the puppy shows 'some aggression' when someone else approaches or tries to take her is quite different.
This puppy is saying "I will decide who is and isn't allowed to touch or interact with me" and that is a definite 'no-no'. It could well lead to some serious behavior/domination/aggression problems as the pup matures.
People often don't recognize aggression or domination in a smaller dog as easily as they do in larger ones. If this was a Rottweiler puppy for example, most likely you would have already realized what this behavior was suggesting.
It's neither healthy or safe to allow the pup to have this kind of control. Imagine if your niece were to get sick and be hospitalized, go on vacation, move away to college or one of any hundred or so scenarios where this pup/dog will need to be separated from her.
If no-one else is 'allowed' to handle her, what would happen?
The puppy won't love your niece any less because it's encouraged to welcome attention and care from other members of the family. In fact, in the long run it will make for a happier, more confident dog. And this is something it's important to start working on now, while she's a young puppy and ready to learn.
Don't tolerate any kind of shows of aggression when others approach her. A very firm "NO" is required when any growling is shown. Continue to handle, pet, pick up or whatever this puppy, if she goes as far as to snap or nip, again a very loud and firm "no' and even a hand wrapped gently around her little muzzle to show that you mean business.
I would also suggest that other members of the family take it in turns to feed, walk and generally care for this pup. She needs to realize that everyone in the household is higher-up on the 'totem pole' and that she depends on them for her basic needs. It's vital that she not be allowed to feel that she's in charge.
I'd also recommend puppy training classes and lots of socialization for this puppy to help her feel comfortable around different dogs, people and places.
Your niece needn't worry that all of this will make her puppy love her any less, that won't happen. Dogs are very intelligent and they instinctively bond with 'their' human/s and know who is their best friend.
The relationship between the two of them has many, many years to go and getting this problem sorted out now won't weaken that bond in any way. In fact, by showing her puppy exactly what she expects of her, and clearly demonstrating that she is the 'alpha' (one in control) and has control over how her puppy behaves, your niece will actually be strengthening that relationship and commanding respect from her pup.
Best of luck with her, hope it all works out well.