Curious behaviour with "order by random()"

I wanted to generate some test data based on a subset of rows in a table,
thus used ORDER BY RANDOM(). I was surprised to see that if RANDOM() is
used in ORDER BY it cannot also be used in column names: the same value is
returned.

Compare the output of:

select random(), random();

which will return 2 separate random values with:

select random(), random() order by random();

which returns two of the same values (and the same value is also used in
order by). While I use 9.6, I got the same results on db fiddle with 13.0:
https://www.db-fiddle.com/f/hNofvnT44izEUmyPyEoWh4/0

What gives? Does using RANDOM() as an ORDER BY somehow turn it into per-row
stable rather than volatile?