Most holidays have, for most people, their "main event" (for lack of a better term) on the Eve before, rather than the Day in Sweden. This includes for example Christmas Eve (the 24th), Midsummer's Eve, Easter etc.
Why, and when did this start? Is it the same reason for all holidays or is it just coincidence?
To add to OP's question, the same holds true for Denmark (and, for Christmas Eve, at least, Germany). Is this a remnant of the Viking Age (or other similarly "Scandinavian" period of history), or is it a more collectively Germanic phenomenon?
This is no coincidence and you can find the same in whole Europe. It is based on old way to count days used till end of middle ages, some countries like Greece followed it even longer.
Nowadays, we consider midnight to be the beginning of a new day. It is also called "German time". It was accepted as new progressive scientific method to count time.
In middle ages, however, a new day started after sunset/with first visible stars. It is so called "Italian time". It came from christian tradition, which itself took it from judaism. In christian tradition, new day starts with vespers, circa half an hour after sunset.
Old holiday celebrations in much of Europe still traditionally follow the old custom and celebrations start on eve of the actual day.