I think it's just a question of personality. Eowyn has a tomboy-ish personality, and it would have been the same had she grown up in a house full of women. The fact that she grew up with a bunch of men just gave her an incentive to show that she was still better than them, even if she was a woman.
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But Melkor also was there, and he came to the house of Fëanor, and there he slew Finwë King of the Noldor before his doors, and spilled the first blood in the Blessed Realm; for Finwë alone had not fled from the horror of the Dark.
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