Your daughter is lucky to have such a concerned and loving dad. I think you have the right idea: It may be that your daughter is one of those kids who is just born highly sensitive. She hasn’t yet learned how to put up a little protective emotional armor. She’s not even 4. Generally she’s happy. She just can’t stand it when she feels big feelings.
I’m glad you aren’t critical of her. She needs help learning how to manage feelings that overwhelm her. Chances are she can’t answer your questions because she doesn’t have words for what she feels. I suggest that rather than quizzing her, you start teaching her some basic coping tools. You can help her learn to take some big slow breaths when she has big feelings. Hold her in your lap and breathe with her. Or you could help her learn to count to 10 as a way to calm down. Or you could encourage her to imagine a protective bubble around her. Be creative with her. Ask her what she thinks would help her calm down.
You want to slowly withdraw the long holding. Otherwise, she’ll learn that a great way to get a long hug is to cry. But please don’t withdraw it before you teach her some other skills. We wouldn’t want her to learn that people abandon her when she unable to soothe herself. Do make sure she gets lots and lots of hugs when she is being her happy self.
I wish you well.