The bug
The box displayed behind text that shows up when hovering over items, things in Statistics, certain buttons and other situations cannot be retextured.
This may be considered a feature request, but it seems strange that something with somewhat intricate detailing around the edges would be code-generated rather than texture-based, as opposed to a box of solid color like was used in 2010-2011, especially given 23w31a's changes to UI texture handling (custom nine-slices, etc.)
How to fix
Introduce a texture file and accompanying mcmeta file for this box.
Linked issues
relates to 1
Attachments
Comments 2

This is now possible in 24w36a.
Can confirm that it can't be modified directly off of a built in texture. Though, it can actually be modified with shaders.
Link may be relevant, not sure: https://github.com/Godlander/tooltip (just an example I've found before, I did not make this)