You can make use of the "Viewport Color" property of each object, since this property can be accessed in any shader. 

See my answer here for more information : https://blender.stackexchange.com/a/163699/86891

In your animation node tree, you need to add an "Object Attribute Output" set to "color" :

[![enter image description here][1]][1]

Then access it in your shader :

[![enter image description here][2]][2]

These two objects share the same shader, but I am modifying the left one's color in my animation nodes tree :

[![enter image description here][3]][3]

In order to change each individual object's color, use this setup :

[![enter image description here][4]][4]

Don't forget to tweak the "Evaluate Fallof" node to output a list instead of just one value. You need to click on the button on the right of the "location" field in this node. Notice how it is blue tinted.
Result :

[![enter image description here][5]][5]


  [1]: https://i.sstatic.net/muPIN.png
  [2]: https://i.sstatic.net/aau5c.png
  [3]: https://i.sstatic.net/QP1DV.gif
  [4]: https://i.sstatic.net/7Bgvg.png
  [5]: https://i.sstatic.net/qvkET.gif