Database
[{words:{wordOne:["Cat","Dog","Fish"],wordTwo:["Red","Blue","Green"],wordThree:["Square","Cirecle","Triangle"]}}]
Query
db.collection.aggregate([{$project:{words:{$reduce:{input:{$objectToArray:"$words"},initialValue:[],in:{$concatArrays:["$$value",{$map:{input:"$$this.v",as:"w",in:{key:"$$this.k",word:"$$w"}}}]}}}}},{$unwind:"$words"},{$replaceWith:"$words"},{$sample:{size:1}}])