The "birth certificate": I assume Cassidy is not Kailey's mother, but how does the MC's name end up on the birth certificate? In my country, the father needs to register the child and declare he is the father, so in that case the MC must know about her existance. Otherwise, the mother would need to register the child but then the father could not be recorded. From a legal point of view, I don't think the mother can declare that a certain man is the father on her own.
"In the United States, there is no national (federal) birth registry, as you might see in other nations, such as the United Kingdom. Instead, birth certificates are issued by the states, which are obligated under law to report annual vital statistics data to the federal government."
[...]
"Within each state, the management of birth certificates might be further decentralized, with data collected and certificates issued at the county or municipal level. Birth data is submitted to the state, county, or municipality by parents, doctors, midwives, and hospitals, typically via paper or electronic forms."
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Basically, the rules & regulations for birth certificates in the USA vary wildly.
States, counties, or municipalities (i.e. cities/towns/etc) may issue birth certificates.
Parents, doctors, midwives, or hospitals may fill out the birth certificate.
This is terrible for consistency, but great for writers.
In the case of Kailey's
(*cough*original*/cough*) birth certificate, Kailey's mother
most likely told a nurse or other hospital employee what to write down on the form.
note: If you think that's bizarre, take a look at marriage laws in the USA. You would think that both parties in a marriage are required to sign/agree/consent to the marriage, but depending on the state regulations you might be wrong.