The correct option is D would
The given sentence states a past habit. The father used to tell stories to the narrator every night ("daily"), hence it was a habit in the past, as that time the narrator was young. Hence the modal must also refer to a past habit.
Option D: 'Would' is a modal, which in this context, refers to a past
habit: 'telling stories daily in the night when I was young'. Hence option D is correct.
Option A: 'May' refers to a possibility of an action that is yet to happen, whereas the action has already happened.
Option B: Similarly, 'might' also refers to an action that is yet to happen. It is the past tense of 'may', in the sense that when 'may' is used in direct speech, it changes to 'might' in the indirect speech. This modal does not refer to a past habit.
Option C: 'Can' refers to an ability, but it does not express a tense. For example, 'I can solve this question'. 'Can' does not indicate a past habit.
Hence options A, B and C are incorrect.