How well did Aragorn know 'old Bombadil'?
In "Strider" Aragorn reveals that he overheard the hobbits' parting conversation with Tom, referring to him as "old Bombadil".  To me, this implies that Aragorn knows somewhat of Tom, since in the first place Tom does not appear particularly old.
"Never ask an Elf for advice, for he will say both 'Yes' and 'No'." - Frodo Baggins to Gildor Inglorion
It's easy to read a lot into that scene. For example, do you really think Bombadil wasn't aware of Aragorn hiding, if he was close enough to overhear whole sentences?

That said, notice that no evil spirits came to the Old Forest during the conflict with Angmar. The Old Forest could have been turned into a lesser Mirkwood - and we have Bombadil to thank that it wasn't. Elrond would have been perfectly aware of Bombadil even in the Second Age, and definitely during the Angmar wars. And we can assume Elrond would have told Aragorn enough to justify that scene.
If the Dunedain of Arnor had forgotten about Bombadil Aragorn would have had plenty of opportunity to learn about him in Rivendell.

But Tom wasn't secretive. He spent time in Bree. He also communicated with Gildor's company - they asked him to watch for Frodo and his companions.

I suspect Tom was well-known to many people outside the Shire, and even a few inside it (including Farmer Maggot).

Of course, Tolkien revised some of Tom's backstory so one must be careful about drawing upon everything written about him.

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