The fact that you are wielding two weapons just increases your armor TN, but to get extra attacks, be it main or off-hand attacks, you need the Extra Attack maneuver or the rank 3 bushi school ability, which is a different ability for each school.
Its covered in their two-weapon abilities. Though, they should have explained it better on the page 141 topic regarding two-weapon fighting and multiple attacks.
The Mirumoto Bushi School focuses on the mechanical benefits of
wielding two weapons simultaneously, both in forgiving the penalties
associated with doing so and improving the benefi ts. It also
increases the samurai’s Armor TN and grants him additional attacks at
an increased rate due to the availability of his weapons.
Meaning that their rank 3 ability is granted because of this two-weapon training, otherwise they wouldnt get an extra attack. Well, all bushi schools get it, but the lore explanation is because of their two-weapon training.
Rank 1: Way of the Dragon
When wielding a katana in your main hand and a wakizashi in your off
hand, you suffer no penalties of any kind for dual wielding, and you
gain a bonus of your School Rank to your Armor TN (This is cumulative
with the normal bonus for wielding two weapons). Additionally, when
you are targeted with a spell, you may raise or lower the TN of that
spell's Spellcasting Roll by 5.
So they take absolutely no penalties when dual-wielding, but nowhere says they can make two attacks in one action, meaning they can't do that without raising the attack roll 5 times, which is highly unlikely to happen at rank 1, as the amount of raises you can make are limited by your Void Ring.
If they could, by rank 3 when they would be able to attack as a simple action, they would attack four times in a round.
However, take a look at the Spinning Blades Kata (pg 260 core), which the mirumoto bushi have access, it allows you to make the Extra Attack maneuver with only 3 raises instead of 5, but that attack must be with your secondary weapon and you cannot raise for increased damage.