Yes, it always did, right from the start. The patch you saw was not authentic.
Each of NYC's 5 burroughs are represented in the logo, there's a church spire for Brooklyn, the "burrough of churches," (I never knew that's what they called Brooklyn) and the bridge represents NYC's connection to the suburbs in NJ and Long Island.
Under "history" on the Mets' website, you'll find the history of the logo.
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