The correct option is
C both
Option A - 'All' means 'every'. It is used for more than one items and not just two. For example, 'All the people loved the movie.' Since 'both' is more suitable, we will not use 'all'. As the sentence describes just two movies, we can't use 'all' here. Thus option A is incorrect.
Option B - 'Either' means 'anyone of the options'. For example, 'I want either of these dresses.' The question asks which movie does the person like, the answer can't be 'either of these', as there has to be a specific name of the movie. Hence, we can't use 'either' here. Thus option B is incorrect.
Option D - 'Any' means 'one of something'. For example, 'I don't have any choice.' As the sentence doesn't explain choice, we can't use 'any'. Thus option D is incorrect.
Option C - 'Both' is used to explain 'two'. For example, 'Both of these dresses look good on you.' As the given sentence explains that the person likes both the movies, we can use 'both' here. Thus option C is the correct answer.
"Which movie did you enjoy more, The Ice Age or Home Alone?" "I enjoyed both of these movies."