Quirrell presumably identified it the same way that Harry & Co originally identified it:
"They're not birds!" Harry said suddenly. "They're keys! Winged keys —
look carefully. So that must mean..." he looked around the chamber while
the other two squinted up at the flock of keys. "... yes — look!
Broomsticks! We've got to catch the key to the door!"
"But there are hundreds of them!"
Ron examined the lock on the door.
"We're looking for a big, old-fashioned one — probably silver, like the
They each seized a broomstick and kicked off into the air, soaring into
the midst of the cloud of keys.
Harry Potter and the Sorcerer's Stone, chapter 16: "Through the Trapdoor" (emphasis mine)
So just because Harry, Ron, and Hermione had a short cut (the damaged wing), doesn't mean that Quirrel couldn't have identified it the hard way.
As for how he got it, I assume he used the broomsticks, same as Harry & Co — that is what they were there for, after all.