I can tell about one aspect of it, but that would be a baby/unborn.
A birth mother can decide to give away the child/allow adoption if she cannot take care of the baby.
She will contact a accredited agency. From them she will get portfolios of "parents in waiting", screened, tested, weighed and measured, willing to adopt. She can opt to choose the adoptive parents, make sure she has a good feeling about them.
When agreed adoptive parents will take care about medical bills and birth costs. After the meds are out of the birth mothers system ( 72 hours after birth ) she will sign the child up to adoption and kind of a pre adotion foster care at adoptive parents. All handled by agency, social workers, lawyers and in court.
After 6 months, visits by social workers at foster/adoptive parents, other hoops to jump, the adoption is finalized by a court, a new birth certificate is made ( Adoptive parents named as parents in there ), the case is sealed and only the adopted child can have it opened at 18/21yrs.
Source:Been there, done that, 12years ago in SanAntonio, TX.
With a "older" child I am not sure. Court would at least have tried to find the birth mother. If not found and no legal ties to Azel I am not sure if she would have gotten custody and after the search for birth mother, would have been able to go through the adoption.
Fathers will alone? Don`t know. Considering the financial situation.. unlikely.
Maybe if she was MCs "godmother"?
An option here would be that birth mother signed over custody over to Azel and calling it adoption was just an easy way to explain for the involved people?
Could that be a cop out from Patreon rules? Not adoptive mother but custodian? After MC is of legal age and his birth mother will mot likely be coming back into his life the legal responsibility and relationship of Azel over/with MC would end