Is there a specific reason why February has 28 days and not some other month?
The choice to use February as the month in which to add a leap day came from Ancient Roman times. At this time, the Romans measured their calendar off the phases of the moon, which is often a highly unpredictable and imprecise method of measurement. Due to this, there were often extra days at the end of the year that had to be added on to make up for the miscalculations, which in Roman times was February.
Under the reign of Julius Caesar, the calendar system was reformed and began to measure dates and months off the sun, which is far more predictable. This meant that days were far more structured, but a leap day was still needed every four years, which was placed in February as was tradition.