The government is not telling you which headphones you can wear. They are saying that these particular headphones work well enough as a hearing aid that it is ok that market them as such. This protects you from quacks that claim their device is a hearing aid but that doesn’t actually work.

Nobody is telling anyone what kind of headphones they're allowed to wear. They do, however, tell _companies_ that they can't claim their product has medical benefits without proving (to some kind of standard) that the product is safe to use, and does what it claims to do. This system was put in place after businesses spent decades scamming the public with "medicine" that didn't do what it claimed to do and, in many cases, was also poisonous.