Gulf of America just makes more sense. Calling this the "Gulf of Mexico" would be like calling The Black Sea "The Gulf of Turkey".
Since you brought up Turkey, they renamed themselves recently. Practically a lot of people just can't keep up every countries nuances and just use the old names. It's fine. It seems like a lot of people still use Bombay over Mumbai too ¯\_(ツ)_/¯
Names are complicated! Germany/Deutschland, Holland/The Netherlands etc.
Interesting that Wikipedia keeps “Turkey”: https://en.m.wikipedia.org/wiki/Turkey
They also keep Gulf of Mexico.
As a programmer: localization is a PITA.