Are operators also considered to be keys?
"$push" is a key in the JSON context. Don't forget everything in mongo (queries, projections, documents, method parameters) are represented as JSON objects. So, although $push is an operator for Mongo, it is still a key on the JSON object it belongs to.

JSON's specification demands key names to be surrounded by double quotes, although the Mongo shell, which is actually a Javascript interpreter, is more flexible and allows key names (field names and operators) without them, as the language does.