When dealing with custody issues the first thing you must determine is who has jurisdiction. Jurisdiction is a State's power to make decisions over a certain issue and can be a tricky thing. When it comes to custody, states look to something called the UCCJEA, or Uniform Child Custody Jurisdiction and Enforcement Act to determine jurisdiction. Jurisdiction cannot be consented to by the parties. So this is not a matter of what two parents agree to, it is a matter of determining which state has the power to make decisions under the UCCJEA.
The general rule in psychotherapist-patient relationships is that the content of the therapy sessions are confidential - in other words, a privilege exists that allows the therapist to not have to submit themselves to the general legal rule that a person should testify to what they know. "Privileges" are given to a variety of professions in order to protect some societal interest. For example, with therapists, the privilege exists so that an individual will feel free to communicate their inner most thoughts, concerns, feelings without the threat of that coming back to harm them in the future.
Under Florida Statute 744.301 a Mother of a child born out of wedlock (unmarried) is the natural guardian of the child. As such, she has ALL of the rights to that child. NOTHING changes this as a fact for unmarried dads EXCEPT filing a Petition for Paternity and requesting that the Court grants you Parental Responsibility (decision making rights) and Timesharing ("custody"). The name of the Petition - Petition for Paternity - is confusing. It leads dad's - and moms - to believe that it is related to DNA testing or who the biological dad is. This is not an accurate understanding - we simply call "custody" actions for unmarried couples "paternity actions".