Hi Brian! SCAG reads:
"Starting at 2nd level, you can invoke a secret elven magic called the Bladesong, provided that you aren't wearing medium or heavy a rmor or using a shield. It graces you with supernatural speed, agility, and focus.
You can use a bonus action to start the Bladesong, which lasts for 1 minute. It ends early if you are incapacitated, if you don medium or heavy armor or a shield, or if you use two hands to make an attack with a weapon. You can also dismiss the Bladesong at any time you choose (no action required)."
You'll note there is nothing explicit about having a hand free. Nor anything explicit about dual wielding or holding two weapons. Nor anything explicit about types of weapons. Nor anything explicit about number of attacks. The only weapon related rules we get are "if you use two hands to make an attack with a weapon". Note that that explicitly states "two hands" with "a weapon". That's two hands with a single weapon. This could be a two handed weapon used with two hands, a versatile weapon used with two hands, or for some reason a one handed weapon held with two hands. This rule is defined not by weapon property related rules, but by how you choose to use a weapon. When dual wielding, nothing about this rule is violated.
In case you don't believe my interpretation, let Mr. Jeremy Crawford make the decision for you. These are his rules after all.
Hope you can learn to live in a world filled with dual wielding Bladesingers!
This is only if you are not going to use the SCAG melee weapon attack cantrips. If you are planning on being more melee attack focus then it provides that one additional attack with an off-hand using a bonus action. So, while I do acknowledge that it is an option, I don't see it as better option and that we will have multiple people choosing it over say someone utilizing the SCAG cantrips instead. Just MHO