I am just surprised since this is the only class ability changing the standard AC formula by adding an extra ability modifier that does not use the stacking-proof phrase. Seems like a curiosity to me.
You have this completely and utterly backwards.
You call this a class ability that changes the standard AC formula. It isn't one.
It doesn't use "the stacking-proof phrase" because it isn't one of those abilities.
In fact, you're wrong even about "the stacking-proof phrase". You need to see that phrase as setting an AC first and foremost, and being "stacking-proof" only second. This ties into a fundamental rule of this edition, on page 14 of the PHB:
Some spells and elass features give you a different
way to calculate your AC. If you have multiple features
that give you different ways to calculate your AC,you
choose which one to use.
It is this rule I am referring to when I say "there's two ways of phrasing it" and "abilities that set your AC. The rules say you must choose one and only one of these."
Things like armor (the rule for no armor on page 14 as well as the rules for light, medium and heavy armor respectively in the Equipment chapter) and things like the Barbarian, Monk and Dragon Sorcerer abilities among others are of this kind.
They all set your AC, and you need to choose which one you want to use.
But not all abilities are like that.
The other kind of ability as regards AC is the one giving a bonus to AC. That includes everything from the Shield spell via a Ring of Protection to, yes, the Bladesinger ability.
The first kind of ability is phrased one, distinct, way. The second kind of ability is phrased another,
distinctly different, way.
And since the Bladesinger ability is phrased exactly right for the second kind of ability, it is the second kind of ability.
No mistakes. Nothing uncertain. Everything says it is fully intentional.
Look Mirtek. I understand you came to SCAG with the strong belief the Bladesinger ability would be like the Barbarian's or the Dragon Sorcerer's.
You say it's strange the way it didn't use "the stacking-proof phrase". That's another way of saying you're surprised it didn't use the correct phrase.
But it did.
What was wrong was your assumptions.
It did use the correct phrase, not just the one you were expecting.
In fact, until you question your assumptions, nothing I say will ever make you change your mind. And until you do, you will never understand why the Bladesinger phrasing isn't strange and isn't a mistake.