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.

To be fair, in the case of hearing aids you are both in the right.

Excessive regulation has created oligopolies and kept prices high in the US. The OTC hearing aid category is meant to help. Before that, low-cost devices tended to remain niche.

OTOH the regulation(s) were introduced due to blatant sales of substandard devices, esp in the 1970s. A high-amplification device runs the risk of further damaging your hearing. Many hearing aid users are vulnerable elderly.

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.