Strange MongoDB $setIntersection behaviour - mongodb

Question

I want to query a match between records in my db based on certain tags. The match would be calculated based on a formula and the intersection of the tags. Now, even querying the intersection doesn't work...always. Sometimes it does, sometimes it doesn't. In my example, if I change the displayName attribute to something else (add or remove one character, the query works. In its current state (for demo purposes) it doesn't as it does not deliver the one intersection match for the last doc with id 3.
https://mongoplayground.net/p/KAYPoV29RFO
That's my query:
db.collection.aggregate([
{
$match: {
"_id": "1"
}
},
{
"$lookup": {
from: "collection",
let: {
"criteria": "$tags"
},
pipeline: [
{
$project: {
"match": {
$setIntersection: [
"$tags",
"$$criteria"
]
},
}
}
],
as: "result"
}
},
{
$project: {
"tags": 0
}
},
])
Here is the example data (simplified):
[
{ "_id": "1", "tags": [{ "_id": "a", "displayName": "a", "level": 1}, {"_id": "b", "displayName": "b", "level": 2}, {"_id": "c", "displayName": "c", "level": 3}]},
{"_id": "2", "tags": [{"_id": "a", "displayName": "a", "level": 1}, {"_id": "b", "displayName": "b", "level": 2}]},
{"_id": "3", "tags": [{"_id": "a", "displayName": "a", "level": 1}, {"_id": "d", "displayName": "d", "level": 4}]}
]
and the result as it is: (expected is three matches for id 1, 2 matches for id 2 and one for the last id. However, the last result has 0 elements in the intersection result. Again, when i change "displayName" to "displayNam" or "displayNames" (obviously in all docs), it give the correct result...
[{
"_id": "1", "result": [
{"_id": "1", "match": [{"_id": "a", "displayName": "a", "level": 1}, {"_id": "b", "displayName": "b", "level": 2},{"_id": "c","displayName": "c","level": 3}]},
{"_id": "2", "match": [{"_id": "a", "displayName": "a", "level": 1}, {"_id": "b", "displayName": "b", "level": 2}]},
{"_id": "3","match": [*here should be the match to _id: "a", but it's not (always) there*]}
]
}]
Does anyone have an idea what I am missing here?