"Under God" is a religious statement. How could it be a cultural or historic reference? I guess you could argue that it is cultural and historic, but what about religion isn't cultural or historic? I would say that it is both religious, cultural, and historic. However, it is mainly religious.
The phrase "under God" was added to the Pledge of Allegiance during the 1940s and 1950s. It is often classed as ceremonial deism, along with references to God on our money. People who call it ceremonial deism claim that "under God" is repeated so often that the phrase has lost any real significance. If we allow this point to stand, do we also then say that the whole of the pledge has no real significance? If so, then why do we use it? U.S. Supreme Court cases tell us that, clearly, there is at least a portion of our citizenry who feels that the pledge and this particular phrase have significance. Words have meaning, and we shouldn't pretend otherwise.