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If you have just one instance of a luxury and you sell it, you won't gain its benefits as long as you are selling it.
Yes it stays, but the yield bonus is not always useful because settling set your city center at a minimum of 2/1.
The best example to test for yourself is to look at a grassland hills with stone on it. Grassland hills produces 2 food 1 production. Stone adds +1 production for a total of 2 food, 2 production. If you settle on this tile you will still have 2 food, 2 production in that city center. A city center's base yield is 2 food, 1 production, so you can see the bonus resource is kept.
Other examples like settling on deer won't work out as well, because while the deer resource is kept, the woods are cleared (losing 1 production from the tile).
When settling on a luxury you will have this luxury so it counts towards your amenities and you can sell it if you want.
A bonus ressource is different. They are only bonuses after all. Some bonus ressources are not worth settling as has been explained by others above. Some are well worth settling in some circumstances, such as rice, bananas and copper. For rice and bananas for example you'll get +1 food, for copper +1 gold.
Do duplicate resources not stack in amenities? There’s no reason to hold on to 2 marble, because the 2nd doesn’t benefit you?
To add to this. Each continent will have only 4 different types of luxuries on it. To get others you need to explore other continents. This is where trading will usually come into play. Unless you are lucky enough to settle on the boundary between two continents, then you will have access to much more variety.