"My daughters are your daughters."
Translation:ילדותיי הן ילדותייךָ.
You're right in my opinion. It should either be ילדותיךָ or ילדותייךְ (masculine or feminine). But using vowels without nikkud doesn't have strict rules, so it's not completely wrong to write with double 'י here for the masculine conjugation.
A copula in Hebrew is usually required between two nouns. However, between a noun and an adjective you can generally drop it, so:
My daughters are happy = ילדותיי שמחות
Though it's understood that ילדותיי would be referring to someones daughters, it would be properly translated as girls. So shouldn't the correct translation be בנותיי הן בנותיך?