The bug
All sizes of chest and large chest, as well as barrels, ender chests and minecarts with chests, point to textures/gui/container/generic_54.png for drawing the inventory, making it impossible to texture them individually.
This is inconsistent with:
furnaces, blast furnaces and smokers, which all use different texture files despite appearing identical
shulker boxes, which have an identical looking inventory to single chests/trapped chests, ender chests, barrels and minecarts with chests, but are handled by a dedicated texture file
How to fix
Introduce new, dedicated files for each of these inventory types.
Linked issues
is duplicated by 1
relates to 2
Comments 7

Droppers and Dispensers also use the same texture file
Hope this is getting fixed soon..

Can confirm in 1.20.5 / 1.20.6

Boats with chests and the two christmas variations of the chest/large chest also use the same texture (generic_54.png).

Another idea would be that different shulker box colors use their own texture file as well as the chipped and damaged anvil. This would allow resourcepack creators to apply a different ui texture based on the blockstate of the container.

Can confirm in 24w18a

Horses, donkeys, llamas, camel and mules also all use the same texture file for their UI.
Can confirm.