Illustrator not saving or exporting assigned color profiles

I noticed, that each time when I export anything from Illustrator as SVG it kept reminding me, that this file doesn't have any embedded color profile (I'm aware that SVG is by default in sRGB - it's just about the popup). This popup shows up at several other instances.

I then always proceed to assign the right color profile, which I already did several times at this point and which is of course also my current working space. But this doesn't change anything: The Illustrator file as well as the exported files still don't have any embedded color profile, no mater what I try. 

The main issue seems the actual AI file itself, not the export. It just won't save the assigned color profile and thus not export it. 

What I tried so far: 

• Assign color profile, save and close
• Create new document 
• Change to another document color mode 
• Change to another document color space
• Reset color space settings
• Reinstalling Illustrator
   

Specs: 

• Windows 11 23H2
• Illustrator Version 29.2.1.
• Dell XPS 17 1910, Intel Core i7-11800H, 64GB RAM, NVIDIA GeForce RTX 3060