"God gave him to us..." "But I heard that storks delivered them!" "No...it was god." "Okay!" "...that was surprisingly diplomatic of y-" "I wasn't kidding, dammit." "....What."
Well, you see Jeane D'rc Alter Santa Lily, when a mommy and a daddy really piss off an omnipotent wish-granting device with their rom com antics and don't watch their words carefully...
As someone that doesn't follow the Fate franchise, I can only say: Seriously, what the fuck is going on in these series!?
Simply put, Jalter tried to one up Saber Alter on Christmas... and used Gilgamesh's potion of youth to try and do it, hence the Lily (younger) part. She specifically wanted to out play Saber Alter at being Santa, since Alter had mercilessly mocked her the previous Christmas over all the variants that Artoria has, up to taunting her that vanilla Jeanne was going to get a variant before the Alter would. So, she ended up with the Santa bit. And she was already Jeane Alter, so you end up with such an incredibly unwieldy name that even the game itself pokes fun at it.
I'm amused that JAlter is like that prolbematic and rebellious middle child whilst Regular Jeanne is the big sis. Jalter Lily for course fits the bill as the youngest.