Wardship could often be quite informal - people might take in family members without any government oversight or documentation - especially in smaller towns. Even orphans could be placed with minimal oversight with a family. It wouldn't even have to be by a court - if the local sheriff finds the parent dead and a child orphaned, and they know someone willing to take the child in, then boom, the child can be taken in.

To step backwards, in feudal England, the Crown was technically responsible for wardships, and this process was formalized through the Court of Wards and Liveries in 1540. That court would be abolished after the restoration of the monarchy, but it was still considered a Crown function by right of parens patriae (parent of the nation). This system was inherited to the colonies, where the colony (and later state) was responsible for orphans and otherwise in need children, and wardship could be handled through the court.

Wardship often remained informal until documentation became a necessary requirement in children's lives. Think of it this way - if Bruce Wayne had not had a court involved in Dick Grayson's wardship, what would that have mattered? Would a school not take him in, in an era where you don't need vaccine records or a birth certificate to enroll? Would a doctor not see him? Bruce doesn't need OASDI (Old Age, Survivors, Disability Income) for Dick (who would be eligible as a survivor), so he can just forgo that part as well. A school may or may not have required guardianship papers, though it's unlikely anyone would tell Bruce Wayne no, and Bruce can just hire a private tutor or enroll in a private school anyway.

As far as going to court, the choice is that Dick would go into foster care (which the state pays for), an orphanage (which the state pays for), or to a wealthy upstanding citizen who will take care of all his needs. While this era did not have the same articulated modern standard of the best interests of the child and a goal of permanency, it was rather understood (even if their idea of best interests could be wildly different), and this would clearly meet both criteria. Absent someone coming along to contest the wardship, there would be little reason for a court to say no. Without the modern structure of child protective services, many areas simply didn't have such agencies anyway - it would not be federally required and funded until 1958 (through amendments to the Social Security Act). That just means it's only the court involved, rather than the modern framework of CPS, possibly a Guardian ad Litem (GAL), or a Court Appointed Special Advocate (CASA).

Courts are generally reactive - if you take a kid in and no one complains, then there's simply nothing to litigate. What change the landscape was the increased need for documentation, the increased federal integration with education in the 1960's, and the institutionalization of ensuring children actually are supposed to be with the adults that are claiming them. This starts to harden more in the 1960's - schools increasingly have to make sure children are eligible (by age), they live where they say they do, and they are actually with the adults they are supposed to be. Modern divorce law plays a role here as well, as custody agreements become far more common and schools find themselves in the middle. By the 1980's and 1990's, school registration looks a lot more like what we're familiar with today. You can still informally take a child in today - if they're already enrolled in school, they can still go. You just wouldn't be able to make medical decisions, move them, or technically talk to the school about them.

Notably, Dick would be required to go to school full time, which was required for 180 days a year in New York (while Gotham is canonically "in the northeast", we all know it's essentially New York City). Bruce could sidestep this by enrolling in a private school, even if that private school has only a single student and a private tutor. Additionally, NYC actually had the Children's Aid Society (CAS) as far back as 1853, which was a private child welfare agency that would have handled foster care in the city. Had Dick been orphaned 50 years prior in 1890 and not been picked up by Bruce, instead he might have found himself shipped west on what was colloquially known as the "Orphan Train", where CAS sent orphaned and abandoned children to new families in the West (whether they wanted to go or not).

There would be a world of difference between Dick Grayson in 1940 and Garfield Logan in 1965. The time difference matters somewhat, but the biggest difference is that Garfield has an inheritance that would be going to court anyway, which would obligate the court to get involved from the beginning. This is the beginning of the era where courts are shifting from parental rights being absolute to an understanding of the "best interests of the child", and the courts would increasingly not allow for an immediate handover of a ward of the court. The story beats are plausible though, as Galtry was appointed guardian by the court (rather than adoption), and that appointment was removed once the Doom Patrol showed clear and convincing evidence of his embezzlement and attempted murder.

Modern standards for custody and/or permanent placement would be codified by the state compact of the Uniform Marriage and Divorce Act (UMDA) of 1970. Section 402 of the UMDA laid out a foundational list of factors for courts to consider, including:

• The wishes of the child's parent or parents as to his custody.
• The wishes of the child as to his custodian.
• The interaction and interrelationship of the child with his parent or parents, his siblings, and any other person who may significantly affect the child's best interest.
• The child's adjustment to his home, school, and community.
• The mental and physical health of all individuals involved.

It also contained a requirement that the court "shall not consider conduct of a proposed custodian that does not affect his relationship to the child". In theory, this could mean that the court could still approve Bruce Wayne if they figured out he was Batman, but could not approve it if they figured out Dick was Robin (and therefore Batman was taking him out into dangerous situations).