No, two-weapon bladesingers fit the rules, fit the spirit, and are intended by the rules.
Fit the rules? It
can be interpreted that way, but "[Bladesong] ends early if [...] you use two hands to make an attack with a weapon," is extremely ambiguous phrasing. It could mean that it ends if you attack with a two handed weapon or with a versatile weapon, but that's not the only interpretation here.
Fit the spirit and intent? Eh... I think it's a huge leap to say that you know the mind of the author. Presumably when they wrote the class, they read through what the class did previously. Is TWF consistent with the class historically?
From the 2e Complete Book of Elves:
Bladesingers rarely use weapons other than swords. With only rare exceptions, they never learn two-handed fighting styles. They may not use shields or two-handed weapons, for these interfere with the bladesong. Even when they have run out of spells, Bladesingers will fight one-handed or with both hands on their single-handed weapon. To do otherwise negates all bladesinging bonuses.
From the 3e Tome & Blood:
Bladesong Style: When wielding a longsword in one hand (and nothing in the other), the bladesinger gains a dodge bonus to AC equal to his Intelligence modifier.
3.5e Complete Warrior's Bladesinger uses the same language, and the additional features Song of Celerity and Song of Fury both include the same "when wielding a longsword in one hand (and nothing in the other)" language.
So, is TWF consistent with the class historically?
No. Not at all.
However, even if you allow TWF, it's not exactly a good idea to do it. 5e still requires that spellcasters have a hand free to cast any spell with a somatic component, and that includes both booming blade and green-flame blade. It also includes shield, absorb elements, and counterspell. Even if your DM allows TWF, doing so basically eliminates your ability to cast spells. Welcome to fightertown!
