I have never been able to find any birth certificates for Emillie, Fernando, Consuelo or Natividad Iznaga Clement using Iznaga, Isnaga, Ysnaga or Yznaga. You say Ysnaga was born in NYC but I think they lived in Orange, NJ.
I have a letter from October 14, 1860 which already mentions Emillie so April 30, 1860 is more likely her birth date. As to the other dates, Emillie was single and may have changed her DOB on passports as a woman's vanity. Their cemetery memorial at Woodlawn only show death dates.
With respect to spelling, there is no "Y" in Basque alphabet, "Y" was used as capital "I" to make it more visible. Also in the 1700's and 1800's it was very common for words beginning with “I” to be written with a “Y” both in common words (Yglesia, Ysla) and Names (Ygnacio, Ybarra ...). In the 1800's in America and in many other countries, Y and I were often interchangeable, as the alphabet was not yet officially 26 letters, and so Y, J, I all were basically used for the same thing as was U for V, thus why a W became shown as two V's joined together.
Although letters C, Ç, Q, V, W, and Y are not used
in writing traditional Basque-language words, they were
included in the Basque alphabet for writing words borrowed
from other languages that do use them.1
Y - Spanish
In the Spanish language, Y was used as a word-initial form of I that was more visible. (German has used J in a similar way.)