No, you can't

You can use your action to make a melee spell attack with the fiery blade. On a hit, the target takes 3d6 fire damage. — PHB p. 242

The rub is that it describes it as a melee spell attack, vs. a melee attack. As such its in the same category as Inflict Wounds and other touch spells as a distinct action vs. a regular Attack action.

Extra Attack only triggers when you make the Attack action

Even if you interpret melee spell attacks as being one and the same as melee attacks, the simple fact remains that Extra Attack only triggers when the Attack action is taken.

In this particular case, the magicalness of the damage matters not.

The Barbarian has resistance to piercing/slashing/bludgeoning damage regardless of whether or not it it's from a magical weapon. Thus they have resistance to the damage from Cloud of Daggers.

However, if you did encounter someone who had resistance to slashing damage from non-magical weapons, that one is up in the air. It's probably magical since the source is from a spell, but that is not certain. Most likely, the reason why it's not spelled out clearly is that it's painfully obvious, these count as magic weapons. However, it's definitely something where you'd need to talk to your DM about it as they may choose to rule that they are, in fact, not magical.