Is $lookup from MongoDB slower than a join in a SQL database, with indexes involve?

Working with Data
,
1 
db.inventory.aggregate( [
   {
     $lookup:
       {
         from: "order",
         localField: "_id",
         foreignField: "item_id",
         as: "inventory_docs"
       }
  }
] )

The $lookup is joining based on the item_id field, which is indexed.
If 100,000 documents pass through this $lookup, it increased the query time to 4X than without this $lookup.

Given that the $lookup is indexed, it is unexpected that the query will be slower by 4X. I was expecting a marginal increase in query time.

Is this also the case for SQL databases? Will an indexed join increase query time by 4X?

explain document:

    {
      "$lookup": {
        "from": "order",
        "as": "inventory_docs",
        "localField": "_id",
        "foreignField": "item_id",
        "let": {},
        "pipeline": [
          {
            "$project": {
              "_id": 1
            }
          }
        ]
      },
      "totalDocsExamined": 0,
      "totalKeysExamined": 100008,
      "collectionScans": 0,
      "indexesUsed": [
        "_id_"
      ],
      "nReturned": 100008,
      "executionTimeMillisEstimate": 18801
    }