When you have to explain how your product isn't a rebadged version of something else, that's usually a bad sign to begin with, but let's see what Fiat has to say on the issue.
Fiat Australia explained to CarAdvice that the 124 Spider / MX-5 was a "joint development", rather than just switching badges. Pat Dougherty, Fiat Australia boss, continued by saying the Fiat version comes with "different sheet metal, different powertrains, different look and feel." He also pointed out that Fiat designers did not see the MX-5 while coming up with the design for the 124 Spider.
Fiat Chrysler Australia's Group communication & media strategist, Andrew Chesterton, later added that the 124 Spider has a different engine, gearbox, and body.
Fiat brand and chief marketing officer of FCA, Oliver Francois, admitted that there are many similarities between the two.
I would say it's a spectrum, so how much do you think the 124 Spider is a rebadged MX-5?