It is normal for all kids to act like "Spoiled brats" if they can get away with it. That's been a given since the beginning of time, especially at age 3 1/2, and especially if they get ignored for it. There are a million "reasons".
If it gets too much for you, or gets worse instead of passing, none of my kids (5, 3, and 2) have any of these behaviors and are very sweet and kind, thanks to this parenting style: Back to Basics Discipline by Janet Campbell Matson.
There is an actual list of 7 things in it that create a recipe fr raising a nice kid.
I was raised this way as well and had a happy loving childhood with respect for my parents and praise for being mature and kind since I can remember. I was rarely disciplined and don't remember it.
No one would ever call my kids spoiled brats. That's a parental responsibility, and you do have control. The key is needing almost no discipline because you are effective, which maintains a very positive and happy home, rather than endlessly implementing "mild" discipline which enable kids to spend hours in wrong behavior, strengthening the habit.
You may want to read about the book on Amazon just to scan. It's non angry, not what you think, and really works quickly so you can get back to enjoying your daughter's good traits all the time rather than dealing with this any longer.
My oldest is now 5, and they say you have to be a parent not a friend, but she's so mature and sweet, she really is my friend thanks to early discipline. She hasn't needed any discipline other than calm verbal explanation since she was 3. My 3 year old son is also sweet and respectful and my 2 year old ex rager is "getting there" on her good behavior after some trying times, but we never gave up and ignored her.
Your daughter sounds very very smart. The more effective your are, the quicker she will "get it". She'll develop a lot of confidence once she chooses to act well, and the good behavior will multiply itself.