The "bank" is likely undercontract as the loans servicer and paid a fee to administer a mortgage they underwrote and/or originated. The 3rd party is the investor which can run from fannie mae or freddie mac to any number of private institutions nationwide (a hedge fund, a contract holder for a CDO/CMO obligation, an individual).
They use a private title-like system to track this ownership and it is not connected to the public title system we use for title insurance and fee simple ownership rights. There is some legal challenges as to whether or not this is legal and whether or not the rights assigned to these third party investors are valid based on real estate law. These, lets call them obligations, are using federal contracts law to push aside state and local real estate law in order to defend the systems they have put in place to insure mortages from default. Some people feel this is what has crippled owners attempting loan modifications and short sales.
The fact you can not determine who the 3rd party is, is not surprising. As enterprising attorneys with deep pocketed clients begin to chip away at this profitable system (that is made billionaires of some and bankrupted AIG and Lehman Brothers, in the secodary CDO/CMO market), clarity will increase. While this won't help you in the short term, what you might try is to find a real estate attorney informed on the CDO/CMO secondary market and see if your home comes up an a search.