Personally, I wouldn't say that refusing to put shoes on at the end of a nice trip for 2 minutes is much at all. Lots and lots of children will take longer than that, if not begging to stay, then crying, or just delaying tactics.
She doesn't always find crying gets her way if nothing was done about the cake being taken. And yes, it's not the crime of the century, but it's also not nice. It was hers, and I don't think being upset about it is unreasonable.
Personally with the ipad thing I'd have stepped in, said "P you had a long play with it, it'll be here again another time." and put it away for the time to say goodbye. Then your dd could have come to say goodbye at the car, and it would probably have been nicer for the ones leaving, not feeling they were leaving in the middle of something they wanted to do.
IME 6 year olds are quick to exploit anything that allows them to get their own way, they will be remarkably persistent even in behaviours that don't work tbh. In your niece's situation crying works for her so she will continue
I would say that when she is in your care then you need to be clear and consistent with her, children are remarkably adaptable and able to understand that different rules and behaviours are required in different environments. As long as you are consistent you should be able to sort this out during the time you look after her. With respect to when in her company with other adults, I think you need to advocate for your daughter when something is unfair, otherwise you have to ignore the behaviour, unless you feel your brother would welcome your advice.
Your house, your rules. Depending on how close you are with your brother, could you talk to him? It sounds like he is finding her behaviour tiresome as well and may just need that extra push to be more assertive himself. It may not change her behaviour much but it may be fairer for everyone else around her.
Personally I think people are digging themselves holes when they feel unable to say no and stand their ground. Sure, kids will whine but such is life...
If you are looking after her one day a week I think your rules go, unless they have discussed discipline with you and asked you to do differently. She's old enough to know that different homes have different rules and to change her behaviour accordingly.
When her parents are there you are just going to have to bite your tongue sorry.
I have two nieces, I'll call them P & C. P is nearly 6 and C is 3. They are both lovely girls and play nicely well most of the time with my DD (5).
P is a very bright girl and always has to be first and always has to win, she also has a bit of a problem with sharing. If she doesn't win or can't be first she cries and refuses to carry on with what they are doing. If I'm watching them I won't tolerate it and ignore. However, if my parents or her parents are there they all pander to her and make a fuss about why she's crying and try to 'make things better' by giving in to her.
She completely over reacts to the simplest of things, C took something off her plate (a bit of a cake) and she just went into complete meltdown. My SIL and B just kept saying oh P stop crying over and over again. They also didn't reprimand C for taking something that wasn't hers (but its cake so not crime of the century)!
P and DD were playing on an Ipad, P had been using it and my DD was just watching and hadn't had a go. P had to leave and put shoes on. She refused to and eventually gave in, this took 2 minutes. She then told SIL my DD was refusing to 'let her have a go' on the Ipad, this is blatantly untrue and poor DD had only had it for minutes.
Cue hysterics, refusing to move, screaming, everyone saying there there P, you can have a go, poor DD is just cast aside and P yet again gets her own way.
My brother finally for once in his life manned up and told her to get in the car, she refused, he repeated and she said no (she's 6 FFS)! SIL said leave her with me and refused to back him up. My parents and her are crowded around P making a fuss and trying to console. My DD is looking on in amazement minus the Ipad!
She has also been known to literally lay on the floor and chuck a massive tantrum in front of other people and its cringeworthy. C is now starting to do the same.
My daughter is no angel, but she only cries if she's hurt or I've told her off.
So how do I handly this behaviour, they clearly don't want to do anything about it and are happy for her to cry and get her own way. BUT I look after her one day a week and were also going away for the weekend, which will consist of everyone tip toeing round her and me biting my tongue!
Do I shut up and put up and let DD be pushed aside or make a comment a take the consequences?