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The following is the method one should use to assign UVs to an Shave Hair primitive. The key step is step 4.

1. Create a primitive.
2. Create a projection (Property->Texture Projection)
3. Create the hair (Hair->From Selection).
4. Transfer primitive UVs to the hair (Hair->Transfer Map and pick the projection from the emitter primitive)
5. Having hair selected, assign a Maxwell Layered Material to it.
6. Add a MaxwellTexture and assign it to MaxwellBSDF Reflectance 0.
7. Add a texture to MaxwellTexture and set Channel Type to "Use Index" and Channel Index to "1".

Here's in pictures:

In this example you are using Softimage hair (shave), not ICE hair, as is mentioned in the title. Unless there is something I've overlooked...

Is there a way to do this with ICE and pointclouds/strands?

Thanks for the correction. ICE pointclouds are exported only as particles/instances atm. Will add support for ICE pointcloud hair in the next update.