a diode drop can be between 0.25 (lowest i have ever seen them commercially) to 1.1V for some of the really cheap ones, so 0.66 seems reasonable,
The beep function you are hearing is called the continuity buzzer, generally used to check that 2 points are on the same wire, now the threshold where that buzzer will sound should be specced in the manual, it just means the voltage across the 2 points is under that threshold,
now as for a diode that reads the same way in both directions, it is likely caused by you measuring it "In-Circuit" and there may well be another diode in anti-parrellel somewhere else on the board, if you lift one of the diodes legs, you may find it only conducts in one direction.
It could even be a resistor in parrellel, as all the diode check function does is output a fixed current and report the voltage, a resistor could just as well drop that voltage for the test current before the diode gets a chance to turn on
In general a diode will fail either shorted (common in zeners), or open (common in conventional) and rarely is a difference of 0.72 vs 0.66 much of a concern