The problem you raise is very mysterious.
I list below what I did, although without a real answer, just a conjecture.
As you say, "File version" is only displayed for .dll, never for .ax,
although .ax stands for a dll that implements a codec.
I have renamed an .ax to .dll, and to my surprise found that "File version"
was now displayed, where it wasn't before, both in Properties and in the pop-up.
Next stop was in the registry, trying to compare
HKEY_CLASSES_ROOT\.ax. Another surprise :
HKEY_CLASSES_ROOT\.ax does not exist.
Ah, ha! I immediately exported
HKEY_CLASSES_ROOT\.dll, replaced all occurrences of
ax, re-imported back into the registry and verified the existence of
HKEY_CLASSES_ROOT\.ax, now identical to
Result: Absolutely nothing. Explorer still refuses to display "File version" for .ax.
One more : To try to force Explore to display "File version",
right-click its column-headers, choose "More..." and check
"File version" . Result: Version is not displayed for an .ax unless it is renamed to
.dll or .exe.
The only conclusion I can offer, is that the display of "File version" is built
into the Explorer program itself for well-known executable file-extensions,
which are only .dll and .exe.
When making up this list, Microsoft unfortunately totally forgot about .ax,
and also all about .scr (dll used for screen saver).
(All testing was done under Windows 7.)