Count number of unique raster pixels in polygon using ArcMap?

Focusing on your main goal, simply compute a zonal majority of the value grid, using the polygons for the zones.

If you still want a count of each unique value within each polygon, provide the polygons with unique identifiers, convert them to a grid (having the same origin and cellsize as the value grid), and combine the two.

These operations require the Spatial Analyst extension.