Problems with import of QGIS generated PDF or SVG files in Adobe Illustrator

I have some issue with importing PDF or SVG files generated in QGIS 3.16 Hannover (with the option to export everything as vectors, no compression, no optimization for smaller file size, incl. all layers, attributes etc.)

In this particular case, I used a digital terrain model to generate a layer of 19 different contour polygons from 0 to 4500 meters in 250 m steps with different shades of grey. In QGIS 3.16 everything appears to be just as I want it. I use the "new print layout" option to generate a map and exporte to SVG or PDF. If I just open the GeoPDF file in a standard PDF reader it looks like this, which is what I want (it’s a random section of the original PDF).

However, importing and opening the same file in Adobe Illustrator gives me something completely unusable:

The same goes for the SVG files, just some unrecognizable garbage.

Any ideas what is going on and how to import a svg/pdf file into Adobe Illustrator with the layer and attributes correctly incorporated? I.e. in this case the 19 different contour polygons with altitude and color info, so that I can continue processing it in AI.