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I first came across it in a manga, in which office ladies discuss what their boss' girlfriend would be like, and one of the guesses is:
きっとすごい派手な女性とか？ オトナーな感じの？
My dictionaries turned up nothing, and searching for the term on the internet, I found オトナーな下着、オトナーなかばん、オトナーなケーキ... I just couldn't figure out a pattern; I thought it was "mature" at first, but a "mature cake" makes no sense. So what does it mean? How come dictionaries haven't registered it?
オトナー is a play on 大人{おとな}, a noun literally meaning "adult", or "grown-up". It is used as an adjective as well to mean "mature", "for adults", "like an adult", "give off an adultish vibe".
「大人な下着」is underwear, especially women's underwear, that is suggestive, possibly see-through or laced. I have also heard women describe black and red underwear as 大人っぽい.
If you describe someone as 「大人っぽい」, especially if said of a teen, the image conjured up is professional, office-lady-like. In your quoted line, the person rumored to give off a vibe that's オトナーな感じ is not a teen, but the idea is the same: a working professional, hence 「派手な女性」
Please also see @naruto's explanation of the different nuances of 「大人の」 and 「大人な」:
Perhaps it's worth mentioning the nuance between 大人の and 大人な. The former is neutral and standard, whereas the latter is slangy and has stronger emphasis on matureness (often sexiness as well). 大人の下着 neutrally refers to underwear for adults, but 大人な下着 tends to refer to suggestive ones. 大人の女性 is neutrally 'adult woman', whereas 大人な女性 is a woman with a typically "adult" atmosphere.
