For one thing, they have Boromir immediately recognize Aragorn's name, as if everybody in Gondor knew that Aragorn was Isildur's heir, they just didn't know where he was.
PJ slips a little on portraying some of the depth of history here, I think. It's been like well over a thousand years since there's been any contact between the Rangers and Gondor and since the last King disappeared in Gondor. So when Aragorn pops up as a candidate for the throne, it would be like a claimant for the present-day British throne who's descended from the pre-Norman Anglo-Saxon Kings. Aragorn does an amazing job of convincing all these fractious aristocrats to whole-heartedly accept him as their king.
LEGOLAS:
This is no mere ranger. He is Aragorn, son of Arathorn. You owe him your allegiance.
BOROMIR:
Aragorn? This... is Isildur's heir?
LEGOLAS:
And heir to the throne of Gondor.
ARAGORN:
Havo dad Legolas (Sit down Legolas)
BOROMIR:
Gondor has no king. Gondor needs no king.
Well, he is Aragorn II. So, maybe (and it's a big maybe) the name Aragorn did ring a bell. Or, as Stewards, they were aware of the names of the kings. (Again, just supposition) So, they knew that Arathorn had a son named Aragorn. They are familiar with the name, but not the face.