With the example result and sample documents, what I understand you want is that for each input documents, you want to find the unique list of the 2nd elements of the Departments array.
You extract the 2nd element with a $set stage like:
set_second_element = { $set : {
second_element : { $arrayElemAt : [ "$Departments" , 1 ] }
} }
Then you $group by using second_element as the group _id like:
group_second_element = { $group : {
_id : "$second_element.name" ,
department : { $first : "$second_element" }
} }
The aggregation pipeline [ set_second_element , group_second_element ] will produce
{ _id: 'Home Audio & Speakers',
department:
{ _id: '63c0faf4752a6a7cfd16960c',
name: 'Home Audio & Speakers',
parentId: '63c0faf4752a6a7cfd169602' } }
{ _id: 'Data Storage',
department:
{ _id: '63c0faf4752a6a7cfd16996c',
name: 'Data Storage',
parentId: '63c0faf4752a6a7cfd169957' } }
Which matches the data of the expected result. I leave as an exercise to the reader a final cosmetic $project to get the data into the exact expected format.
When you redact documents for publishing please ensure we can cut-n-paste them and insert them into our system without having us to edit them. The last sample document was not terminated correctly. The "_"id:“Id” produce duplicate errors.