Does a master's research project (not a thesis) count as a "publication" for PhD applications? My master's in mathematical finance doesn't have a "thesis" exactly, but it has a research project where we helped a bank develop a model to help with credit risk by identifying which variables were more important in credit risk management (based on the particular data set we had which was the financial information, such as net income or return on assets, of the clients of the bank). This wasn't some report at the end of an internship, so our research project is somewhere in the middle of an internship and a thesis, I think.

My coauthors and I printed out a document that had an abstract, review of related literature, scope, methodology, etc which was then put in hard bounds and is now sitting on the shelves in the math department of my university.

We gave the first/a mock presentation to the risk management department and then the second/a real presentation to our "thesis" advisers.

The document is about the application of some articles, journals, papers, etc that led to us developing a model.

In a PhD application form, I'm asked for any publishing. Does the "thesis" count as a "publication" ? If so, is the "publisher" the university?

The options for type of publication are:

"Books/Book Chapter, Conference Paper, Journal Publications, Others"

The options for status of publication are:

"Published, Not yet - accepted in press, Not yet - Under review, Not yet - writing in progress"
If it's printed in your institution's library it is fair to say that it was published by your university. You may want to explicitly list it as a technical report in your bibliography to clarify the type of publication that it is (your audience will want to know if your publications are peer reviewed or not). 

Under the list you have here it would be an "Other" that is "Published".