For some reason "_id: 0" replica member never has tag "serviceName" : "my-cluster-name" while other replica set members have it.
Here's how it looks:
Some other info:
1. externalNodes do not get this tag
2. if we issue rs.stepDown() and primary steps down it is not added/removed anywhere
3. cfg replica members do not get it currently
So basically currently it looks like inconsistency and we should also investigate why it was added in the first place (maybe it's just obsolete and should be removed).