No. Eidolons do not have rich parents, nor were they raised by the church.
EDIT: I mean, I guess your Eidolon could take the "Extra Traits" feat, but that would be just weird and I think most GMs would rule most traits as completely unthematic for such an outsider.
Traits are an optional rule, and how many you get is up to the DM in the first place!
So it wouldn't be crazy to ask if you can apply traits to your Eidelon instead, but you need to ask your DM, not us.
Character traits are only for player characters.
Third Paragraph on the page.
You can argue that Eidolons can take the "Additional Traits" feat as described in the next section, as Roberta says. If one of my players tried tried that I'm my table, personally, I'd either laugh at them, politely ask them to leave my gaming group, or throw a book at them depending upon how I was feeling*. Because I 100% agree with LazarX; eidolons are class features, not characters. While an eidolon can mechanically benefit from Character Traits, they do not fit ANY of the traits flavor at all.
* I loathe the summoner class in its entirety.
Wow. What a bunch of, well I don't know, but it surprises me that you wouldn't allow an eidolon to benefit from a perfectly viable feat.
There's nothing that says eidolons don't have their own pasts and lives apart from the summoner.
For all we know, they were normal people too before becoming aeons--er, eidolons. Perhaps some of those personality traits show through.
Let them choose traits in place of/ as feats. I love summoners and eidolons. Fun FTW. ;)
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If the player wants to spend one of his Eidolon's feats on Additional Traits, and picks two that could fit the theme of the Eidolon, I'd be fine with that. (For instance, someone whose Eidolon is their 'phantom twin' who grew up with them in the same region, allowing that Eidolon to start with two of the local / regional / cultural / racial traits might be perfectly appropriate. Someone who summons up a chicken headed snake that's a manifestation of their own inner turmoil, not so much. Someone whose eidolon is described as an incarnate outsider of some sort, a protean or fiend or celestial, could well have 'traits' of it's own, carrying over from it's previous outsider-y existence, or even a *pre* outsider existence as a mortal!)
But, starting traits? No. No more than a familiar or animal companion or leadership cohort or other random NPC would get such a thing.
Barring the Additional Traits feat, traits are pretty much PC-only.