Separation meltdowns in small children are not a sign that something is wrong with your child.
They are a sign that your child prefers you over other caregivers.
As their primary attachment figure, you function as the place they feel the most secure, so for a few young years (usually between 9mo-3.5/4 years old), they won't want to separate from you, especially in situations where there are no other family members or close familiar relationships.
Though this is emotionally laborious for us as caregivers, their clinging is a natural part of a secure attachment relationship in early childhood.
Try using separation rituals to help your children feel some control in the process and have a positive moment to look forward to in the separation process.