It's just a fashion convention, and a mostly outmoded one at that.
Saying they're "not allowed" is drastically overstating the case. :)

The custom was that one should wear white shoes only during the summer,
which officially starts on Memorial Day (holiday commemorating service
people killed in war), which is at the end of May, and ends on Labor Day,
which is at the beginning of September. In some circles, it was seen as
a faux pas (perhaps literally :)) to wear white shoes in the wrong season.

I've always thought it was a weird and non-useful tradition, so I've
never paid much attention to it. But I don't think it was something
observed much by my generation (baby boom) - my mother used to remark
about it, at first seriously, but later, more facetiously.

Joyce

Ummm.... "not ALLOWED"????  SFAIK, that's one of the
outmoded fashion dos-and-don'ts that most people observe
only if they feel like it.  (They got around the one about
not wearing white clothing - other than shirts - by
inventing a shade called "winter white".)

There may have been a practical reason for the white shoe
edict, simply in the interests of keeping them reasonably
clean.  Cold weather can set in fairly early in some parts
of the U.S., it tends to rain a lot in the autumn, and back
in the days when coal was the main source of fuel for
heating buildings, rain + soot from coal smoke = very dirty
shoes.

My guess is that the custom originated in a part of the country where autumn
gets chilly and wet, and wearing white shoes is asking for trouble when it's
muddy. I also think it started with shoes and then extended to anything else
white.

Blessed be,
Baha