Colour changes between illustrator and photoshop, why is this happening?

I tested a cut and paste between InDesign 2020 and Photoshop 2020 and the hex values changed. Then tested InDesign 2018 to Photoshop 2020 and the hex values were correct. I don’t have an earlier version of Illustrator to test, but the "bug" may have been introduced with 2020?

I’m seeing the same in CC2020 whether I copy and paste or place the .ai file. If I open the .ai sRGB file in Photoshop, the hex color is unchanged.

First of all, hex is just base 16 notation for RGB numbers. As such, they are subject to standard color management, and numbers are color space specific. The same color in sRGB and Adobe RGB will yield different numbers and vice versa.

Secondly, the embedded document profile will always override your working space. Color settings aren't all that important and there's no reason they need to be synchronized.

All that said, what we have here is a one-step shift in the R and G channels, up and down respectively. That's a very typical result of jpeg compression. So I'll guess that your PS document is a jpeg?

Jpeg compression is very aggressive, and irreversibly destructive. In addition, the color component is much more aggressively compressed than the luminance component. Colored banding and numbers shifting is to be expected.

If you need accurate numbers, you cannot use jpeg.