Different damage types are just gameplay mechanics, influencing a champion's item synergy and their opponent's counter building options. Don't overthink it when it comes to the lore.
Regarding Garen's R, I am 99% sure he can not do anything remotely like that in the lore. If his ultimate did physical damage in-game, then opponents would be able to build armour and largely nullify his damage, making Garen useless as a juggernaut. Magic damage means that the ultimate remains a powerful spell even against an armoured opponent.
Regarding Fiora's W, her sword is just a sword, Demacian Steel or not. If Fiora kills you, the specific cause of death will be blood loss, not magic. Demacians don't like using magic anyway. I'm not sure why the ability does magic damage in-game though; perhaps to remove the black cleaver synergy?