Using QGIS polygonize tool to create discreet slope polygons

Assigning new colors to the grid does not mean that the pixel values are categorized. The raster values remain untouched. The polygonize tool does not know anything about this categorization.

So first you need to reclassify the raster. Check this: Link and do it with the Grass r.reclass tool --> it works perfectly. Once you have the ammount of categories you want you can polygonize the raster.

You could classify your raster trough the raster calculator with a simple formula like this:

("Slopes@1" > 15) + ("Slopes@1" > 30) + ("Slopes@1" > 45) + ("Slopes@1" > 60) + ("Slopes@1" > 100)

Each condition returns a 1 if true, so the raster output will have 0 values for source pixels less or equal to 15, 1 values for source pixels greater than 15 but less or equal to 30, 2 values for source pixels greater than 30 but less or equal to 45, etc.

Then, polygonize that raster.