Both Seasons will display if they overlap, once the season has ended, the Season name will be removed from the Season dropdown.
The current Season (based on the date in Core > Settings > School information > Years & terms) will display first in the dropdown. If there is an overlap of season, you will see both seasons listed in the dropdown until the first season has ended. At this time, it will switch to the next season and the prior season will be removed from the Season dropdown.
If you enter Athletic Clearance for Fall and its Season required and then you try to enter Athletic Clearance for Winter. As long as the Season was selected in the dropdown for the Athletic Clearance it will go with that season.