It is not classified under either. There is no classification for sugarcane. It is a grass. Not all the plant material we eat need to be classified as a fruit or a vegetable. Here is a thumb rule

1. Vegetable: These are certain parts of plants that are consumed by humans as food as part of a savory meal.
2. Fruits: In common language usage, it is the fleshy seed-associated structures of a plant that are sweet or sour, and edible in the raw state

Apart from these we consume the following as per terminology

1. Grain
2. Pulses
3. Nuts
4. Herbs
5. Spices

There are items such as sugarcane, maple syrup, beetle nut leaves to name a few that do not fall in any of those categories

Edit: I meant to type betel leaf and areca nut. My answer was autocorrected to betel nut.