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I think the screening can also be used to determine if the child will go to full day or half day kindergarten, depending on what your district offers.

It's required here for enrollment. It's typically done in preschool though. But if a child doesn't go to preschool, they do it before kindergarten.

This is what's included according to my school district:
Who Needs to be Screened?
All children living in this district who are 3, 4, or 5 years old and are not currently enrolled in Kindergarten NEED to participate in this FREE screening program before entering Kindergarten, or have an equivalent screening from a doctor. Children who attend Head Start or Early Childhood Special Education Services programs may not have to participate in Early Childhood Screening.
Screening assists your child to be ready for Kindergarten. Health and developmental problems or concerns can be identified earlier and help provided sooner; families may be referred to community health and educational services including School Readiness and ECFE.
Screening is free, easy and good for your child.
Early Childhood Screening Includes:
• Immunization records
• Hearing and vision
• Speech and developmental screening • Height and weight
• Summary interview with parents

What Happens During Screening?
The parent attends the entire screening process with their child. Your child will have a developmental progress review by a speech/language clinician. A review of your child’s vision, hearing, height, weight and immunization history will be conducted by one of our health professionals. The parent/s will then participate in a family factors interview to review the screening process and talk about any concerns they may have.

Yes, it is just an assessment of ability. My friend's son did this for K (public school) last week. She said that they do an assessment so they can create a diverse mix in each K class for the fall.

It's a normal thing here, they do it when you enroll the kids. They check for things like being able to count to a certain number and "reading" books in the proper way (holding it so the words face up, turning the pages in the right order, etc).