Aligned index column order
You need to think about your partitions as if they are separate tables.

A clustered index on a partitioned table effectively has the partition number as its first key. For you, with a partition per unique PartitionKey, the clustered index will essentially have that column in the front of it. Or as good as... 

I'd expect having Id first would feel better, in case you don't know the month and need to search all partitions. It'll seek on each partition rather than scan them.