It's normal.
But you must stop asking her to choose things (like which parent will pick her up) - you and her father must be the parents and make the decisions. YOU decide who does what. Today is Daddy's day to pick up, tomorrow is Mommy's. Done. Tonight Mommy puts you to bed with story time and lullabies, and then Daddy comes in for a kiss; tomorrow is the reverse. Done. Set a schedule and try to stick to it. If it doesn't work on a certain day, just tell her matter-of-factly that this is how it's happening today because someone is working or tired or busy or out for the evening.
Kids cannot have everything they want. If they want pancakes and French toast and cookies and stale pretzels for breakfast, it's not going to happen. If they want to pick out $100 in cereal and watermelon at the supermarket, you get to say "no". It is okay, and in fact essential, that kids learn early on that they don't get everything they want, that choices are limited, and that Mommy and Daddy are the bosses. They can cry and fuss and plead and beg, but it doesn't make you change your mind. You say she "needs" both of you to play with her or she "needs" 3 stuffed animals. But she doesn't. She WANTS those things. If she wants to go get 3 stuffed animals from her room, fine. But you don't turn the car around because she only remembered to take 2, and you don't drop everything you are doing so both can play with her. If you don't teach her to understand "no" (without 10 minutes of justification), she will have a miserable time in preschool, kindergarten, and life - there are rules everywhere she goes, and other people besides her have needs and wants.
The longer you let this going on, the harder her habit will be to break. I suggest you put in some structure and limits sooner rather than later. Give her a choice (maybe, if it works) of "Mommy story, Daddy tuck-in/kiss" or "Daddy story, Mommy tuck-in/kiss" but NOT "Mommy and Daddy do everything." And if it becomes a problem, then you and Daddy make the decision and inform her of what it is.
This is the same logic as "Do you want strawberries or blueberries with your yogurt?" vs. "What do you want for breakfast?" Kids that age cannot handle wide open choices, so you limit them while you let them learn from selected options. And you don't give in to needless crying or tantrums, which are also normal at this age. A lot of people refer to the "Terrible Twos" but you are now the parents of what is also known as a "Three-nager" with the same mood swings and nonsense as you will see when she is a teenager! Practice now on how to face attitudes and bad moods!