Closest we have would be something like 'Blood Brother", but that has mostly fallen out of use. But yeah, that is very situational, and not something you would use as an outsider.English doesn't have an equivalent term. Calling someone unrelated to you a brother or sister is sometimes used as a turn of phrase, but only in rather specific contexts. You also would never use the term for other people; you wouldn't say, "how is your sister?" unless you were talking about their actual sister, even if they called that person their sister before.
Since they grew up in the same orphanage, Hana could be called her adoptive sister or her foster sister, but that's stretching the meaning of those terms.
And yeah, that is stretching it a lot, since adoptive or foster sister would imply they were adopted by the same parents, and only Hana was ever adopted. I mean, they might think of each other as sisters, but they have never really talked like that.