Its a phase.
Or has she always been this way?
All I know is, developmentally, all kids/babies go through 'separation anxiety' and it is manifested differently, per age.
Even Kindergarten kids have a hard time separating from the Mom at times. For example.
Maybe, your daughter will benefit from going to Preschool? Just part time, half days. Unless she already does that.
Teachers, are used to this.
And it would help her socially as well and developmentally.
I have 2 kids, and each kid has their own personality and cues.
And at different age stages, a child can become more attached or "clingy" with the Mom. My son, was like that. He was VERY close to me and attached. At certain ages, he became more "clingy." But it was a phase. I knew that. But so, at 4 years old, he began Preschool. He was ready then and adapted just fine. It was a maturity on his part and my knowing him and his readiness etc.
Regardless, a child has to be with someone else, because Mommy has to go somewhere or do things. So, you need to do that.
Even if she screams.
Now: WHAT do the other family members do with her, when she is screaming???? How do they... handle her????
This, also matters.
She needs to feel comfortable with them... and they need to feel they like her despite her emotionality. Having FUN things, or incentives or people being nurturing... can help. And also, talking WITH her, beforehand can help too. So that, she knows what is going to... happen. Some kids, NEED to be forewarned. So that they can get it settled in their head and so "abrupt" changes are not, sudden.
2 years old is hard.
3 years old is harder.
4 years old is also hard.
Because, the "emotions" of a child at these ages, are not even fully developed yet, nor their communication skills, nor are they fully aware of their own feelings or why and they cannot communicate it. But you can teach her... how. ie: when my son was only 3 years old, he could tell me... if he was "grumpy" or "frustrated" or just irritated.
And then per "coping skills".... a parent has to help teach the child that. Children cannot instantly "cope" with things. It is taught. ie: if my son is grumpy, he will go to another room and be by himself. That is HIS way. And he will tell me. And it works, for him and us.
The book "Your Three Year Old," or "Your Four Year Old," or "Your Five Year Old" is a GOOD series. It is simple to read and not a thick book. But it simply explains what a child at this age is like. Although written years ago... it is very pertinent still. Amazon has these books. I have it too. I like it.
Sure, you can do punishments or scold her etc. But along with that... a child needs to be taught, HOW to say her feelings, to know they can say it, and then teach her coping-skills for times when they are feeling frustrated. Kids often get frustrated. So we need to teach them how to... manage... their emotions. Otherwise, they grow up not knowing how nor can they express it, and then they get all pent up and just have hissy fits. Because they do not know how... to say their wants or emotions.
Its hard, but this is a hard age.