There are a number of saints named Elizabeth so you need to be more specific.
The feast of Saint Elizabeth is November 5.
I can find no saint by that name.
We celebrate his feast day because he is Our Lord's Adopted father.
The feast day of St. Elizabeth of Hungary is November 17.
January 4 is Elizabeth Anne Seton's feast day.
The feast of St. Elizabeth ithe mother of John the Baptist is November 5 in the Latin Church.
Elizabeth of Hungary is memorialized on November 17.
On September 29 we celebrate the feast of Saints Gabriel, Michael and Raphael the Archangels
There are many saints named Elizabeth so be more specific.
A feast day is the day we celebrate the life of someone who has been recognized as a saint. In the Roman Catholic Church, days are assigned to certain saints. Usually the saint's day of death (or entrance into glory) is chosen, but often either the day of death is not known or the calendar is crowded and another significant day will be chosen. Many people celebrate the feast day of the saint after whom they were name. For example, someone named Nick might celebrate Dec. 6th, the feast day of St.Nicholas, as their own feast day, too.
Greeks don't traditionally celebrate birthdays. They celebrate their Name Days (the feast day of the saint they are named after).
I have placed links below where you can click to learn how some countries celebrate the feast of Lucy.