The answer to the original is question is no, you're not a rival just bc you play in every sport. You could be in the same district for every sport and not necessarily be a "rival".
When I think of different rivalries, I think of rivalries developed by history (the Soul Bowls), prestige (Carol City/Northwestern/Central), a school coming about to take students away from another (Cypress Bay/Western), neighborhood/proximity (Muck Bowl/Carol City vs Norland/NMB vs North Miami Beach)