Unless you want a girl that grows up believing she can cry and whine to get what she wants, and that material possessions are more important than anything else, you need to stop it now, and tell your parents to stop it as well.
You can't control the fact that the other girl has things that your daughter likes. This girl might not even have more per se, but it may just be that your daughter really admires her in some way and the things this little girl has seem way "cooler" than what she has. But you can start putting your foot down because in the end, it's about the kind of woman you want your daughter to grow up to be, and the kind of values you want her to have. And 3 seems awfully young to already be aware of such things. My daughter is 3 too, but while she notices the things other kids have (like a certain backpack, or Spongebob Crocs) she does not ask me to get them for her too. She doesn't even seem aware that it's an option. It's just not on her radar.
My daughter cries about stuff when she doesn't get her way too but I don't let it wear me down. That's life, get used to it. There will be plenty of things in life that will disappoint them and make them cry and the sooner they learn to deal with them, the better. By giving in to your daughter's crying, you are teaching her that 1. crying will get you what you want and 2. she does not have to deal with any disappointment and negative feelings she might have (and eventually turns to shopping, or food, or whatever to feel better). To some degree, you are, in essence, enabling her. I think of it like when they get vaccines - they are getting small non-harmful doses of something bad that will make them stronger and make them better at dealing with the bigger bad things when they happen.
Time to nip it the bud now - it will be easier now than when she is 13 and won't take no for an answer. Or 23 and already has credit card debt out of control.