[{"date":20200817,"items":[{"name":"item1","values":["val1","val2","val3"]},{"name":"item2","values":["val1","val3","val4","val5"]},{"name":"item3","values":["val1","val3"]}]}]

db.collection.aggregate([{$project:{output:{$map:{input:{$filter:{input:"$items",cond:{$in:["$$this.name",["item1","item2"]]}}},in:"$$this.values"}}}},{"$project":{"output":{$setIntersection:[{$arrayElemAt:["$output",0]},{$arrayElemAt:["$output",1]}]}}}])