Hi, I'd like to help you with your question.
This is a phase that some children go through. It can start in the toddler years and last into adulthood. It all depends on how it is handled by the parents (and other adults in the child's life).
There are a few possibilities as to why your niece is responding this way.
One, this is part of her personality. Right now that seems like a negative. But when you consider how some CEO's are made or athletes get ahead, there are just some people born with a competitive spirit. At your niece's age, she is too immature to understand how to express it, but with the right guidance she will get it right.
Two, your niece is just going through a phase. At her age, she is right on the edge of adolescence. These days, kids mature earlier so you can almost consider her a teenager. Teens are by nature questioning who they are and measuring their self worth against others. They are fragile in terms of how others see them and if they are good or bad at whatever they try. They don't have any experience to fall back on and almost everything is a competition. Plus the are not mature yet to know how to handle the emotions related to losing. So they act out like a child. Until they learn better coping skills, that is all they know.
Three, what are the messages your niece is getting at home? I don't mean to imply they are bad. But she may feel competitive with siblings who are better at things that she is. Or Mom and Dad may be at the top of their field and she feels overwhelmed by expectations. Sometimes parents compare without realizing that they are doing it. All parents make mistakes. And teens are sensitive to most all things.
With any of these possibilities, it just takes time and some support to help her through. Keep talking to her like you have already done. What you did for her was great. She may respond with "I know", but inside she appreciates the support. She also hears more than she lets on. Also, show her how to lose gracefully. You and the other adults can let her see that when you lose, it's no big deal. Laugh about it, poke fun at each other and generally make light of it. Emphasize the fun rather than who wins or loses. Everyone celebrate at the end together.
If you find over time that your niece does not improve and/or gets worse, consider suggesting counseling. In most cases, this will pass over. But it does not hurt to see if there is an underlying reason for her feelings if she is unable to eventually let go and mature.
I hope this helps,