Evidence;
-mentions at the end of the school year that he wins in the end, because he never had to pay rent
-easy access to strong alcohol
-only income seems to be from panhandling and illegal gambling
-residing in a school lax truancy enforcement
-only attends the class full of students who cannot identify him and works to avoid all other contact
-dresses heavily to conceal his body type, like many criminals
-rants like a lunatic
Perhaps.
However he did mention he had parents. They may be paying his way through school, and nothing else. Also, he has his own money. (How do any of these kids get money, anyway? Do they have jobs? Allowances? Does the game ever say?)
A lot of Kenji is hidden from us. We do not know enough about what others know about him to make a judgement on whether he is a student or not. Perhaps he has "special needs" for time and space to himself
like Hanako does.
That Kenji is plainly visible in classrooms from time to time - such as when he was making a sign for the festival - makes me wonder if he's really all that incognito. The teachers probably recognise him as "that weird kid in the thick glasses, coat, and scarf".
Perhaps the thing that convinces me most that Kenji is actually a student is the presence of guards themselves and his manner of dress. Considering his conspicuous clothes, one would think he'd be pretty easy to spot. If he were not a student, wouldn't the guards be able to spot him and haul him away if they did not recognise him as a student?