In modern equal tempered music Dm isn't any sadder than the other minor keys. It's only in older tuning systems (ex https://en.wikipedia.org/wiki/Well_temperament) the different keys had different feelings.
If nothing else instruments play differently in different keys. Horn players like certain b flat, guitarists love g, tuning a stringed instrument higher or lower changes the feel. But I am not convinced 440hz is the same as 500hz etc, on the physiological level.