The procession is the pre-start of the wedding ceremony. This is where all the guests are seated and the wedding party enters. Before any words are spoken, this parade lets everyone get into place, so that the oral part of the ceremony can begin.
The procession has evolved to include rituals and traditions of its own, however the main purpose is to get everyone seated in the room.
If the wedding were a stage play, the procession would be the part where the house lights are dimmed and everyone is seated so that the show can start.
Many wedding traditions are based around heterosexual marriage. As a result, the language used to describe certain roles and events are intrinsically gender based.
For same-sex weddings, it can be useful to use the existing gender based terms to assign roles to the bride, groom, wedding party members, etc. In other words, by designating one partner to stand in the traditionally "male" side and one in the traditionally "female" side. It really doesn't matter who takes what role, but it helps determine an order so there isn't confusion about who enters when and who stands where.
Alternatively, same-sex weddings open up many opportunities to create new wedding traditions. If you are planning on officiating a same-sex wedding, make sure to work closely with the couple to tailor a ceremony that is respectful of their values.