Currently it seems that rasters export without an alpha channel -- they maintain a similar look, but their opacity is set to 100% even if you had the transparency set as something different in Pro. Therefore, you're not able to get the 'original' look of the layer within Illustrator without going back to Pro and doing a new export.
For example, here's my land cover layer in Pro - Multiply with 75% transparency.

The layer looks the same in Illustrator, and it has multiply applied, but the opacity is set to 100% on the group and all sublayers down to the image itself.

It seems the colors were converted to their non-alpha equivalent as they appear in Pro. Please just export the original image with transparency applied.