Does il mio onomastico in Italian stay the same feminine or does it change?

"Il mio onomastico" stays the same, regardless of whether you are male or female. The phrase itself is not gendered.

However, the specific name you are referring to in "il mio onomastico" will determine whether it is masculine or feminine. For example:

* Il mio onomastico è il 25 aprile (My name day is April 25th) - This is the name day of Marco, which is masculine.

* Il mio onomastico è il 15 agosto (My name day is August 15th) - This is the name day of Maria, which is feminine.

So, while the phrase itself doesn't change, the context will reveal the gender of the name day being celebrated.