Only one attack can be modified by a Blade Flourish
There is another Q/A regarding the sorcerer's Metamagic which states:
You can use only one Metamagic option on a spell when you cast it...
There is now a small semantic requirement/argument that "you can use only one metamagic option on a spell" and "you can use only one metamagic option per spell" are equivalent statements.
To me this is true and so the above sentences are equivalent. This sentence structure was ruled in the Sage Advice Compendium to mean that you cannot use the same metamagic option multiple times per casting:
A sorcerer can use one Metamagic option once in the casting of a spell, not the same option more than once...
The Bard's Blade Flourish feature has the same structure stating:
You can use only one Blade Flourish option per turn...
It should follow the precedent set forth by metamagics and the Sage Advice Compendium and so the Blade Flourish clause means that you cannot use the same Blade Flourish option multiple times per turn.
To spell this out more directly:
You can use a metamagic option once per spell casting, and you can't use the same one multiple times in a spell casting.
You can use a Blade Flourish option once per turn, and so you can't use the same one multiple times in a turn.
This is also supported by Jeremy Crawford (though now unofficial, and at best can only be considered rules as intended) here stating:
Blade Flourish can benefit only one attack on each of your turns..
A small note on wording: Blade Flourish does not say that you can use it once per attack because then you could use different Flourishes with each attack.
Only one attack can be modified by one Blade Flourish option per turn.