Is there a reason why you did not use Product Attributes which are stored Automatically ?
If you used Prodct Attributes then it would be automatically displayed on the product page, in the cart and in the order?
What does the DropDown value represent ?
If you want to display the Drop down value in the Cart from a plugin then you need to make an override and new a new razor page.
If you want to store the data you can either make a new table or store it in order.CustomValuesXml