Hi Juhaan,
The mapping works via your implementation. You need to set the 'setCustomerId' parameter to the encrypted id that you are uploading in your implementation, that way it will map the ECID or MID parameter to the encrypted client Id that you mention.
Customer IDs and Authentication States
There is a limited backfill method that can be setup where analytics will export the evar value (if you have it already) and the subsequent visitor id's to the customer attribute system. If you want to set that up you need to open up a ticket with customer care. But, this is just meant to be a stop gap until you get your implementation setup to set the customer Id's real-time.
I hope that helps,
Seth Burke