1570s, "condition of being an orphan," from orphan (n.) + -age. Meaning "home for orphans" is by 1850. Other words for "condition of being an orphan" have included orphanhood (1670s); orphancy (1580s); orphanism (1590s); orphanship (1670s); and Middle English had orphanite "desolation, wretchedness" (mid-15c.). Also in the sense of "home for orphans" were orphan house (1711); orphan-asylum (1796); orphanry (1872).
ORPHANAGE ... is a very incorrect expression for an orphan-home. Fancy a "girlage" for a girl's home. "Orphanry," like pheasantry, diary, aviary, is the proper word, though I believe it is in no dictionary. "Orphanotrophy" is enough to send one off in atrophy -- a word fearful and amazing. "Orphanhood" is a good word, and expresses the state of being an orphan. That the root of the word is Greek, and the affix English, is, I think, immaterial, because the word "orphan" is so thoroughly Anglicised that we are never thinking of [orphanos] when use it. ["Notes and Queries," Jan. 20, 1872]