Q:

(the) project BankABC, BankABC project or the BankABC project?

I struggle with proper usage of articles in the following situation: I have a project for a customer, let's call the customer BankABC. Is the following then correct?

This is the description of the BankABC project. //I guess here BankABC
  acts like an adjective

or

This is the description of project BankABC.

A:

The first is generally better:
"This is the description of the BankABC project."
And you're right, the noun is being used like an adjective.  This is called an "adjectival noun," or a "noun adjunct."  See the following for more:
https://en.wikipedia.org/wiki/Noun_adjunct
But the second version isn't altogether and completely wrong.  As Peter Jennings points out in his comment, project would have to be part of the proper name of the project itself, "Project BankABC."  Perhaps a goverment program teaching bankers how to read?  Lol.  Names like this aren't super common, they're used to sound a little more grand, a little more flamboyant, like something you would see in a James Bond movie, or something a politician would name something grandiose.   

