you're exactly right: the "name day" celebration has its roots in the Catholic celebration of saints' days--so it makes perfect sense that traditionally-Protestant nations like the UK and the US don't celebrate them.

Although yes, originally it was connected to the celebration of one's own patron saint, there is no longer any explicit connection to Christianity. I'm a Wiccan (or, at least, soon-to-be one) and I still celebrate it :D

In fact, here in Mexico, is greeted with "Feliz Día de tu Santo." (lit.: "Happy Day of your Saint."). It even has its own celebratory song! It's not celebrated as it used to be though, it has been falling out of favor over the last decades.