The permanent address question is usually because a lot of international travel insurances will ship you back to your home country if that works out cheaper. Fine print alert: Most international travel insurance policies are what you call “secondary insurance policies” they are sort of an insurance of an insurance.
So if you put down your Swiss address, for a high cost situation, they might decide to “transport” you back to Switzerland for treatment (where like you said you could re-register and get treated for free…) - that’s where your “primary” Swiss coverage kicks in.
This varies in every situation and depends on your country of origin, where you are when you need treatment, the severity, etc. etc. etc.
Putting down your “permanent” address in your international health insurance as Switzerland (a parent or close relatives address) should not affect your tax situation. It’s just where the health insurance folks send their policy letters, and will ship you back to, if they have/want to.