“Golden Visa” is a residence permit scheme. The Golden Visa programs grant investors the right to live in a country in exchange for a qualifying investment. The investment options, the benefits, and the extent of the preferential treatment differ depending on the issuing country.
That’s correct. A residency permit in any country in the Schengen zone allows you to travel freely within the respective countries, subject only to their individual immigration restrictions. ie you cannot spend more than 90 days in any individual country without registering and a residency permit in one country does not allow you to reside in any other country. So a PR card from Greece would allow you to travel to, for examle, 60 days in Italy, 70 days in France, 80 days in Spain etc but you could not move your residence to Spain.
As a practical matter, you could probably get away with spending longer than 90 days in a country as there are no checks, but the legal limit is 90 days.
For tourists wanting to travel long term in Europe this results in the “Schengen Shuffle” where you spend 90 days ‘in’ and then 90 days ‘out’ of the Schengen countries. The 90 day requirement is based in large part on the principal of reciprocity between countries. The US, Canada and other countries only give Europeans a 90 day tourist visa. It’s a kind of ‘you get what you give’ between nations.
Immigration requirements and long term residency rules are set by each individual country. If you want to stay longer than 90 days in a country you have to apply for some type of long term visa, a work visa, a retirement visa etc. Each country sets their own rules. You can obtain a permanent residency card in Greece by buying property there (subject to a number of other requirements) and this allows you to reside for as long as you like in Greece. As a resident of any Schengen country you can visit any other country - you must carry your passport and your residency card. Unless you have an EU passport, you are still subject to the 90 day limit in any country.
As long term residency/immigration is a national affair, you cannot ‘purchase’ a Greek residency and then go and live in France. For immigration and residency requirements you need to search by country as each is a little bit different, ie France’s requirements are extremely bureaucratic and pedantic, while Belgium is a little more relaxed. Requirements vary by both the country you want to reside in and by the passport you hold. https://france-visas.gouv.fr/web/france-visas/long-stay-visa
If you have a PASSPORT from any EU member country, ie you are a citizen of an EU member country, you can move to and live in any other member state at will. You are still required to register your stay with that country for social insurance and health care purposes etc. but you have the right of free movement within the EU.
A residency permit DOES NOT give you the same rights as a PASSPORT.
Visa requirements, whether short term or long term are based on your passport, not your residency.