I was a teacher (in MA) for several eons.
If the kid is an orphan, he's a ward of the state. If he's a ward of the state, he's placed in a foster home or group home. (Until age 18, anyhow.) You just won't find orphans roaming around free - unless he's a runaway, but if he's a runaway, then he's been found, hasn't he? Because he's living with a teacher, a teacher who'd know he can't just take in some kid and pretend he's adopting him.
But the kid's in school, so the school needs to know who the kid's parent or guardian is and who he's living with - along with the paperwork to back it up. This kid can't just live with the teacher because it's not enough that people 'think' the teacher has adopted the kid. And if the teacher is adopting him, there has to be a social worker and/or state official overseeing and approving the whole process. (And the school is kept in the loop throughout.)
Can teachers adopt kids? Sure they can. I knew many teachers who were foster parents. But they had to go through a rigorous process to make sure they were fit to be foster parents, and most of them were married. Many had natural children of their own.
However, if the teacher is a wizard, perhaps he's worked some magic or forged some documentation that will certify that he is indeed the kid's legal guardian or foster parent. But why bother? Why not have the teacher/wizard simply, legally adopt the kid? (Or you as the writer, can say that he was adopted years ago?) Do teachers and their own children and adopted children attend (or work at) the same school? Of course they do.
I know kids slip through the cracks. I know kids get lost, missing, abducted, etc., but you've established a scenario where the kid goes to school and just happens to live with a teacher and apparently that's okay with everyone. That rings false to me. Even if the teacher tried to do it on the sly - get away with it - someone somewhere is going to speak up. Either another teacher is going to start asking questions (it's part of their job) or someone else will: an administrator, counselor, the other kids, or maybe even a parent. Or someone who lives on the same street and thinks something funny is going on.
We caught a thirteen year old girl (Grade 7) living with an older man when I was teaching. How'd she do it? Her mother was on a business trip and had her staying with a girl friend and the friend's family. But the girl, after her mother left, lied to this family and said her mother cancelled the trip, so she had to go home. She didn't go home; she went down the street and stayed with a guy in his 30's. How did we teachers find out? (We didn't find out for three days.)
Her friend, a boy who knew her since early childhood, came to me and said I have something to tell you, Mrs. M., that I'm not supposed to tell anyone but I have to tell someone. Yeah, I was shocked but we called the police, and the mum on her business trip and the family she was supposed to be with and later the girl was sent to a private boarding school.
I like to think teachers try to be vigilant in this area without being unnecessarily intrusive into our students' lives, but it's all part of the job.