If you mean at the same time, obviously Chicago and New York. If you mean just 2 different teams that is a whole different can if worms. This is a partial answer. There have been many cities that have had two MLB teams at the same time.
1) Philadelphia - Athletics (1901-1954), Phillies (1883-current)
2) Boston - Red Sox(1901-current), Braves (1876-1952)
3) St. Louis - Browns (1902-1953), Cardinals (1892-current)
4) New York - Yankees (1903-current), Giants (1885-1957), Dodgers (1890-1957) Currently Yankees and Mets(1962-current)
5) Chicago - White Sox (1901-current), Cubs (1876-current)
6) Los Angeles - Dodgers (1958-current), Angels (1961-1964)
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New York City: Yankees, Mets, Giants, and Jets
Chicago: Cubs, White Sox, and Bears
Philadelphia: Phillies and Eagles
Baltimore: Orioles and Ravens
Washington: Nationals and Redskins
Atlanta: Braves and Falcons
Miami: Marlins and Dolphins
Tampa Bay: Rays and Bucanneers
Pittsburgh: Pirates and Steelers
Cincinnati: Reds and Bengals
Cleveland: Indians and Browns
Detroit: Tigers and Lions
Saint Louis: Cardinals and Rams
Kansas City: Royals and Chiefs
Houston: Astros and Texans
Seattle: Mariners and Seahawks
San Francisco: Giants and 49ers
Oakland: A's and Raiders
San Diego: Padres and Chargers
Minneapolis, Denver, Phoenix, and Dallas are debatable: each has a baseball team that plays near that city, but their team names (Minnesota, Colorado, Arizona, and Texas) identify with a state, not the city.
The New England Patriots do not identify exclusively with Boston, do not play in Boston, and do not even play in a suburb of Boston.