Civilization – Harvesting Resources in Civ6

TL;DR Changes in amenities won't show up until you hit the next turn button.

In Civ 6, each luxury gives you 4 amenities, and extra copies of a particular luxury don't give you more amenities. According to page 50 of the manual:

Only one source of a specific luxury provides amenities, shared with the four neediest cities in your empire.

You don't have precise control over which cities get which amenities, and "neediest" isn't defined anywhere, so I don't know exactly where those amenities are going to go.

In addition, it often takes to UI a long time to update things. In this case, the engine that distributes amenities among your cities only runs after you click end turn, so the UI will definitely not reflect the new one until the next turn.

Here are the bonuses from being a Suzerain as listed in the wiki:

• The city-state follows you into war and peace with other players.
• Your units may enter city-state lands.
• Your Builder units may improve city-state tiles.
• See 3 tiles around the city-state.
• Can pay Gold to temporarily take control of the city-state's military.
• Gain ownership of all the city-state's resources.

• Not listed but you are also give their unique bonus for being their Suzerain. All the city states and their unique bonuses are listed -> here.

You specifically asked about the last one, gaining ownership. This is the same as it was in prior Civ games. By being the suzerain you get the luxury and strategic resources that they have. The luxury ones provide amenities (if you don't already have that resource), and strategic ones can be used for the corresponding units.

In no Civ games including this one do you ever get food/production resources from being the Ally/Suzerain of a city state. They just list that you control all resources, but the excess bonus resources do not give you anything as they are for providing bonus production/food on specific tiles.