MissScarlett: This does indeed turn out to be a spurious Morphy game. Chapter 26 of Lawson's <Paul Morphy: The Pride and Sorrow of Chess> discusses the authenticity of the few disputed games.
<The doubtful Medley game first appeared in the London <Era> of January 2, 1859, as a Medley-M game. Very evidently the "M" stood for Mongredien and not for Morphy, for Mongredien was mentioned in the notes to the game, and it never appeared on any list of Morphy games at the time.> (Kindle ed., ULPress, 2010)
Here's the relevant section of the January 2nd column (p.5):
<We have lately, through the kindness of Mr. Medley, the Honorary Secretary of this club, had several fine games placed at our disposal illustrative of the attack in the Evans Gambit, which we shall from time to time lay before our readers. The members of the London Club now practise their favourite game with much energy and spirit, and Herr Lowenthal has accepted an invitation to be present each Friday evening, and enjoy a contest with Mr. Mongredien, one of its strongest players, during the winter season.>
Then comes the game. Whilst I accept that Lawson is likely correct to equate <Mr. M> with Mongredien, it's not quite as explicit as he proposes. Mongredien's name features in the (above) introduction to the game but not the <notes to the game>, as we now understand the term.
Noteworthy too is that a Morphy game, the second match game with Anderssen, appears in the same column, tagged as <Mr.A.> - <Mr.M.>. Could a conflation of the names be the simple explanation of how this game came to be misattributed to Morphy?