My Sov often marries the female hero assassin and his female children inherit the assassin title and get the doubled-damage for their first attack in a tactical battle.
@cbarchuk -- my guess is that by 'once per season' he means a player turn.
My experience is that my assassin heroes and children get the doubled-damage attack for their first attack (not counter-attack, counter-attacking before they can initiate an attack does not trigger the doubled-damage attack) each tactical battle. If there's multiple tactical battles in a player turn then they get that doubled-damage attack in each tactical battle.
Also, if the doubled-damage attack is blocked or dodged, that counts and the doubled-damage attack is considered used for that tactical battle.
And my Sov has never gotten a doubled-damage attack (when the wife and female kids have the trait) so I think Gene1966 is correct).