I saw somebody use this on twitter, but I had to do a second take because to me it doesn't make sense. If you expand the acronym you get "our President of the United States". It seems redundant if an American says it, since the it is already specified that the president occupies the American presidency. Is a non-American says it then it seems confusing, almost implying they have their own office named the President of the United States.
In the current context it probably implies ownership of responsibility: our national shame, our dark secret, our troubling history supporting fascism.
Also consider that #notmyPOTUS is a hashtag on Twitter, calling out the illegitimacy of a election that awards the office to a candidate who received fewer votes.
In a specific conversation it might mean a favorite: "My POTUS is Abraham Lincoln, who is yours?"
It is grammatically correct.
It is illogical, however, as it would imply that there may be more than one POTUS at any given time (i.e. there's our POTUS, but also your POTUS (a different person), and also their POTUS (yet another fellow or gal).
POTUS is a nickname, not an official acronym. POTUS doesn't appear in the government's style manual. Because it's a nickname, informal usage isn't ungrammatical.
Informally, "the POTUS" and "our POTUS" are only substitutes for "the president" and "our president." Strictly, "the POTUS" and "our POTUS" are like "the Donald" and "our Donald," an emphasized name.