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Eidolons are not creatures that reproduce or have ancestors to hand down an heirloom weapon. If the heirloom process is happening in the imagination of the summoner as you fluff it out above, then he is the one who should have the heirloom trait and only he could get the trait bonus and free proficiency. If the item were truly the heirloom of the eidolon it would have been acquired in its independent life as an outsider and would vanish with it when it returned to that plane.
That said, you could fluff it as a cherished weapon passed down to your eidolon from its mentor. It's too cheesy for me, but if your GM is fine with it, that should cover you.
If I were the GM, I'd allow it like this, but it's strictly homebrew.
The Summoner I would make take the heirloom weapon trait, but, I'd allow him to apply it to his eidelon instead (this would be due to it being 'part of his pscyhe', so he can instead apply it to the Eidelon, but not himself.
The fluff would be :
Erik came from a long line of inquisitors. It was a tradition in his family that the first born received the great great great grandfather's longsword. It was tradition that this happened on the 5th birthday, so the child grew up playing with the sword and was familiar with it.
Erik was not like the rest of his family, he was a loner, and had a vivid imagination. He would go out into the wilds by himself, despite all the scoldings and punishments. He always claimed 'Barbra' protected him, and she use his shiny ancestor sword. No one believed him of course...
Until in his 11th year, his father found him in a clearing, dead wolves all around, and a bloody and scratched warrior woman with glowing green hair and a strange rune on her forehead guarding him. She wielded the ancestral longsword, covered in wolves blood. Erik lay behind her, his ankle turned at an unnatural angle, blood dripping from bites on his legs. As soon as the warrior woman saw who it was, she relaxed and faded from sight, the sword thudding to the grass.