[RPG] Can a bladesinger dual-wield

You can wield & attack with two weapons without any special class feature or feat.

Like most contests, however, some restrictions may apply. From the PHB, p195 (emphasis mine):

Two-Weapon Fighting
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

As long as both weapons are light weapons, you can attack with both at the cost of an action for the first attack, as normal, and then a bonus action for the second attack. The second attack does not receive your ability modifier as a bonus to damage the way attacks normally would.

Fighters & some other classes have an option for their Fighting Style class feature which is also called, confusingly, Two-Weapon Fighting. This feature is not required to attack with two weapons as described above; the benefit it gives is to allow you to add your ability modifier to the second attack's damage.

The Dual Wielder feat is likewise not required to attack with two weapons as described above; it instead provides additional benefits and removes other restrictions/penalties when you choose to do so.

Dual Wielder and the Two-Weapon Fighting style make you better at fighting with a weapon in each hand, but are not required to do so.

RAW: You can use Bladesong while in Wild Shape, and vice versa

You can enter a Bladesong either before entering Wild Shape or while in Wild Shape, as the Wild Shape feature states (emphasis mine):

You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so. However, you can't use any of your special senses, such as darkvision, unless your new form also has that sense.

Jeremy Crawford states in an unofficial tweet that:

Bladesong doesn't require a weapon in hand. Its description would say if it did.

And that a Bladesinger does not need to have his or her weapon in hand in order to utilize the Bladesong ability.

One thing to keep in mind is that you cannot maintain Bladesong if (per SCAG, p. 142):

you don medium or heavy armor or a shield, or if you use two hands to make an attack with a weapon

So keep that in mind when considering what to Wild Shape into.

You can use Bladesong while wildshaped, or can use it prior to wildshaping and still maintain its effect as long as you pick a shape that fits with the rules of using Bladesong.