If you had 2 features that said "once per turn when you hit you can add damage", would you only be able to use one of them?
If the features were the same feature yes; one time per turn. For example if I have a Fighter 1/Rogue 1 and get Nick on a dagger from both classes I can not use Nick twice in one turn because I have it two instances of it which each allow me to make "one" extra attack. When I get to fighter 9/Rogue 1 I also can't use Dagger-Nick from being a Rogue on the same attack I use Dagger-Sap from being a fighter.
But besides this you are also talking about two different versions of Extra Attack and this is covered under multiclassing.
From the 2014 PHB:
If you gain the Extra Attack class feature from more than one class, the features don’t add together.
From the 2024 PHB:
If you gain the Extra Attack feature from more than one class, the features don’t stack.
So the Extra Attack feature from the Bladesinger does not stack with the Extra Attack feature from the Valor Bard or for that matter the higher level features of Fighters.
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