She has a human body model and human height. I propose she is half elf of the Feynriel ilk. If that's not enough for you, immediately after walking away from her to talk to Cole he said this...

I've believed this for awhile; glad to see there's some proof! (I really need to talk to Cole regularly.) I think this would explain why she's not "elf trained"; she has the wrong physique to fit into the armor. Alternatively, she won't put on "elfy" gear. It's a pity it doesn't trigger a cut scene with Sera when you try to equip her with Dalish scout armor and she refuses to wear it. "Arrow in your face" and a stream of other Sera-isms would likely follow!
Heck, they should a number of cut scenes where companions react to the gear you give them. I can imagine the colorful remarks from Vivienne when forced to wear gear that doesn't make her look like a fashion model. Or Blackwall, when receiving a Grey Warden outfit.
Sera can open the locked cells beneath the Chantry in Haven for you if your a mage or a warrior (actually she just automatically loots them) but I just cannot trigger that conversation with her in the Haven pub is anyone else having this problem?
After a few conversations where you build approval with Sera, your inquisitor should go into the basement and try to open all the doors and verify that they're locked. I think only after you do that will it trigger the dialogue choice where you can ask Sera to open the cells. If you've done that and the dialogue option doesn't trigger, maybe you need to build approval with Sera a bit more. In the dialogue options, tell her what she wants to hear...
Her real parents are unknown, but she was taken in my Lady Emmald very early on.
Then she ran away and a Jenny took her in - a Human, mind you.
She was trained by Humans. The armor says "Elf-TRAINED" thus it means TRAINED BY ELVES. Only Solas and Elven Inquisitor meet the requirement.
Check the codex on Sera after she tells your IQ about Lady Emmald; the codex updates to say that Leliana's people cannot verify that a Lady Emmald ever existed in Denerim; that plus the unknown real parents makes me think Sera isn't telling the truth about her pre-Red Jenny life.
As to "elf-trained", "human trained", etc., I think those are just racial categories. I don't think Solas really has any special "elf training." He's nostalgic for ancient Arlathan but has no special sympathy for the Dalish or the city elves. (After the winter palace, he takes offense if you assume he would be upset if things didn't work out for Briala.) Although he seems a bit taller than other elves, he still has the right physique and thus fits into Keeper Armor, etc.