Is a woman's saying "Yes, I have a daughter" or "Yes" when asked about the child she has sponsored considered a double entendre (tawriyah), if she intends that she is raising a daughter, knowing that the child is not her biological daughter and that she is an abandoned infant?

Tawriyah (ambiguity/equivocation) in speech is permissible if there is a need for it. In this case, your statement that the child is your daughter falls under the category of permissible forms of equivocation, because you mean that she is like your daughter in upbringing and care, and to achieve the child's welfare in treatment and affection. This is with the emphasis on the necessity of clarifying the truth when a suitor proposes to her, and that official documents must bear her true name and lineage.

Also, there is no harm in a Muslim being called by any name, or by the name of his nephew's son. The proof of this is the Prophet's (peace be upon him) calling Aisha (may Allah be pleased with her) "Umm Abdullah." Therefore, there is no harm in people calling you "Umm the mentioned girl" (mother of the mentioned girl) even though she is not truly your daughter.