Integer encoding and weighing when one feature consists of more names
That's a good question. This task would be referred to as multi-label encoding. Bascially if a movie belongs to several genres, you one hot-encode each genre and add the vectors.
If there were only 6 genres (horror, romance, action, adventure, comedy, fantasy)
For instance a movie that is horror, action and comedy (The Dead don't Die?):

horror = [1, 0, 0, 0, 0, 0]
action = [0, 0, 1, 0, 0, 0]
comedy = [0, 0, 0, 0, 1, 0]

So a movie that belongs to all three would be encoded as: [1, 0, 1, 0, 1, 0]. A movie that only has a category then its encoding is a one-hot encoded label.
You can perform this task in scikit-learn with a multi label binarizer