How to group objects when each object could go in multiple groups?

by

You basically need a Stream that is made out of a Pair (I choose AbstractMap.SimpleEntry here) that has the left part as a Hobby and right as the Student (could be the other way around, does not matter).

Later just group those based on Hobby (that is a String in your case).

data.stream()
    .flatMap(student -> student.getHobbies().stream().map(hobby -> new SimpleEntry<>(hobby, student)))
    .collect(Collectors.groupingBy(
            Entry::getKey,
            Collectors.mapping(Entry::getValue, Collectors.toList())
));

Entry::getKey being a method reference that gets the key, you could write it as a lambda expression too, if it makes more sense for you:

Collectors.groupingBy(entry -> entry.getKey())

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