I am no expert on European residency law, but, I wouldn't read anything more into that 'requirement to register' that what you see at face value, meaning, if you become resident in a country, you need to register with the local authorities. That could be necessary (from the point of view of the local authorities) for any number of quite innocent reasons, for example, enabling the local authorities to track population numbers in communities in order to ensure that the communities receive federal or regional grant money, etc.

It's also unclear how you would go about registering if you are in fact a bona fide tourist, meaning, someone who keeps moving around the country and does not stay in one place. It is quite possible that the country you want to visit could simply give you a tourist visa longer than the default 'no visa required' 90 day period if you asked them for one. Many countries do exactly that.

Best suggestion I can offer is that you visit the Consular Section of a local Greek embassy and ask them what your obligations, rights, and restrictions are.

You'll have to read your policies to find out. Insurance policies such as this are independent business transactions between you and the insurer, and the number of possible variations are endless. If in doubt, phone or write the insurer(s) and ask them.

Michael