No—"Mr. President" is not just the name of the job preceded by the traditional male English honorific. It's the actual title of the sitting president, the way something like "your majesty" might be used for a monarch. Edit: /u/The_Alaskan's comment reply notes that "Madam President" would be used by a female president, and that a formal decision to that effect was made in advance of the 2016 U.S. general election.
So did this hold true for Dr. Wilson? Yes! From my own study of America during that time period, I recalled seeing Wilson addressed as Mr. President in print, but I wanted to confirm that by finding a verifiable example with a source. The best I've got for now is a reference to a sign carried in the NAACP's 1917 "Silent Protest Parade" that read "Mr. President, why not make America safe for democracy?" Source: Alan Dawley, Changing the World: American Progressives in War and Revolution. Edit again: /u/Lo-Jupiter pointed to additional, earlier examples of Wilson being addressed as Mr. President in this comment reply!
But why is Mr. President the title, and not something else? Initially, after the formation of the young U.S. government, there was a bit of a debate over how to address the president. Some in the early U.S., including the first president and vice president, favored a more exalted title. VP John Adams suggested "his highness," and George Washington preferred "his mightiness." Adams believed the executive was appeared too weak in comparison to the other branches of the new government, and sought to consolidate presidentail power boost the appearance of the president's power by way of a grandiose form of address.
However, there was vehement opposition to this style of address among many other leaders, who felt it was a bit too reminiscent of the monarchy they had all just worked so hard to escape (Thomas Jefferson said an exalted presidential title was "the most superlatively ridiculous thing I ever heard of," and Benjamin Franklin called Adams' stance "absolutely mad"), and ultimately Washington and Adams yielded and accepted the title "Mr. President." Edit a third time: /u/yodatsracist points to a source in this comment reply suggesting that Washington himself never actually went by Mr. President. My sources (listed below) suggest that the title was adopted through common usage because the House of Representatives opposed anything more elevated than that, but don't say whether or not Washington himself ever used it.
I'm not an expert on the American Revolution or its aftermath, and I'd be happy if a real expert could flesh out or correct any aspects of that hurried section of this explanation! Here are the sources I used for my quick study on it:
Edits for clarity, and to add additional info!