No.
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.
Mirumoto Bushi
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.
You prevent this by either killing or banishing the spirit before they can act. Killing is usually the prefered method used by most shadowrunners, but is by far the least efficient, unless the entire team can act before the conjurer does, which could happen. The conjurer has to command his spirit and spend one service so they can force/ask the spirit to use a certain power, unlike telling the spirit "Fight them!".
This means that removing services from a spirit, exactly what the Banishing skill does, is the fastest method of getting rid of spirits. It's a contested roll of Banishing + Magic [Astral] against the spirit's Force (normally around 6 dice), and each net hit means one less service for the conjurer to use. You would be surprised at how good this works.
A lot of people will not recommend the Banishing skill (preferring to Stunbolt them), especially if you treat it as secondary to your other skills (and most do), resulting in checks that only net one or two successes, not enough to fully banish the spirit and leading to frustration. But a good (and possibly Edge-fueled) Banishing check can remove the spirit from the game completely in a single action.
If you know what is coming your way and you realize it's a nasty effect (like Fear, Confusion or Compulsion), it's generally a good idea to use your Edge before you make the roll, so you can take advantage of the exploding 6's. Re-rolling should only be done when odds are in your favor, but the dice are not.
Best Answer
Below are all of the sources I could find to combat fear in the Core, Secrets of the Empire, and Element Books.
Stances
Advantages
Schools & Tattoos
Spells