Can you specify a key for $addToSet in Mongo?

You can qualify your update with a query object that prevents the update if the name is already present in profile_set. In the shell:

db.coll.update(
    {_id: id, 'profile_set.name': {$ne: 'nick'}}, 
    {$push: {profile_set: {'name': 'nick', 'options': 2}}})

So this will only perform the $push for a doc with a matching _id and where there isn't a profile_set element where name is 'nick'.

As of MongoDB 4.2 there is a way to do this using aggregation expressions in update.

For your example case, you would do this:

newSubDocs = [ {'name' : 'matt', 'options' : 0}, {'name' : 'nick', 'options' : 2} ];
db.coll.update( { _id:1 },
[ 
   {$set:  { profile_set:  {$concatArrays: [ 
      "$profile_set",  
      {$filter: {
             input:newSubDocs, 
             cond: {$not: {$in: [ "$$this.name", "$profile_set.name" ]}} 
      }}
   ]}}}
])