I would understood "On był/został wychowany przez swoją babcię" as "He was raised by his own grandma", that is, the grandmother in question is definitely mother of his mother(for example). On the other hand, "On był/został wychowany przez jego babcię", I would understood as there is another person(not mentioned) and grandmother of that another person was the one that raised him… Since those are two different persons, depending on the meaning the course creators wanted here, one of the options shouldn't even be accepted, imho. ;-)

Nevertheless, both are grammatically correct sentences, just with different meaning.

I have a feeling, (but I do not know the rule for this),

that zostać fits better with perfective verbs(dokonane)-wychować

while być is necessary for imperfective verbs(niedokonane), -wychowywać