U.S. carriers generally allow one to fly standby on an earlier flight
than booked (on the same day) if there are open seats, for no
additional charge. The one time I asked about this when traveling on
a flight within Europe the agent politely but firmly said that I could
fly only on the exact flight that I had booked. This was on BA,
flying out of CPH.

Is it general policy within Europe not to allow same-day standbys, or
does it vary from airline to airline or country to country?

Before anyone flames, I'm not arguing one way or the other, just
scoping out the possibilities.

Depends on the airline and the mood of the checkin staff.
Twice I have tried this once with TAP who politely told me to buy another
ticket and once with Flybe who had no problem whatsoever.