Okay, I'm going to try this one more time.
When you DO NOT mouse over the resources shown in the top right corner of the city screen, you will see three columns of resources. All but the left column will have icons for happiness or health next to the resource icons. These bonuses go to ALL YOUR CITIES.
The production bonuses (for having the correct resources to build Wonders, Cathedrals, Walls, etc) also go to all your cities.
When you DO mouse over these resources, you will see a second bonus - food for Pigs, commerce for Wine, hammers for Iron, etc. These bonuses apply to TILES, and therefore only a city working a tile with such a resource gets that bonus.
And no, multiple instances of the same resource do not help you in any way. This is why trade is a good idea - trading away your surplus Wheat for the Wine you do not have any of will lose you nothing while gaining you +1 happiness in all your cities.
EDIT:
Ragnar, the bit you're not getting is that all your cities
do get a bonus - they get the ability to produce certain units, or build certain buildings faster.
IAM, where you and I misunderstand each other is that all your cities do get health as you describe; but if a city is working the deer tile, that tile has +1 food yield because of the deer. Plains deer would have 2F1P; without the deer, it would simply be 1F1P.