Hmmm, I hadn't noticed that detail (I've only seen the film once!!), but I don't find it too surprising. It's clear enough that Father Christmas in Narnia is the same Father Christmas (aka Santa Claus) we know in our world, much though Lewis makes the point that he's not just "fat and jolly" like most of our depictions of him — he makes the children feel "glad, but also solemn". Presumably he's already visited them and left gifts for them every Christmas since they were born, so he knows very well who they are and what their names are (and whether they've been naughty or nice... ). So it's not a stretch for me to believe that he knows why they're in Narnia and that they're about to fulfil the prophecy and become Kings and Queens, not least because he hasn't been able to visit Narnia since the White Witch took hold, until now.

(Don't forget, he also knew all four of J.R.R. Tolkien's children by name and wrote to them personally every year, with their full names on the envelopes, too...  I do always think that's the biggest irony about Tolkien reacting so negatively to Lewis bringing Father Christmas into Narnia!!)

Now there's something I hadn't thought of!  I guess it's just natural to assume Father Christmas is exclusively a Narnian resident in this story, but it would make sense that he, just like Aslan, would also exist in our world- but with the same name in at least one country.  

 So I guess this just brings it back to Aslan apparently not knowing Peter's full name until Mr. Beaver tells him (in the movie, that is). But that's a shortcoming of the movie that everyone has already pointed out- I just thought it weird that the filmmakers saw fit to give Father Christmas more information on the kids than ASLAN HIMSELF. It's especially weird because Aslan greets the kids by name immediately upon meeting them and without having any time for anyone to tell him anything about them, but then turns around and acts like Mr. Beaver had to fill him in on the other details, like Peter's last name and hometown.  

Peter's 'hometown', of course, isn't actually revealed in the books. The starting paragraph of LWW just tells us they were 'sent away from London'. (I think Finchley came from Anna)
The children don't have a surname in this book, but the film makers thought they should.  

Posted by: @coracle

Peter's 'hometown', of course, isn't actually revealed in the books. The starting paragraph of LWW just tells us they were 'sent away from London'. (I think Finchley came from Anna)
The children don't have a surname in this book, but the film makers thought they should.  

  True. This is more of a film-related question, based on a weird choice made in the design of the quiver prop.

The book doesn't have these narrative problems, fortunately!  

Posted by: @the-mad-poet

I guess it's just natural to assume Father Christmas is exclusively a Narnian resident in this story, but it would make sense that he, just like Aslan, would also exist in our world- but with the same name in at least one country.  

It's made clear enough in the book that he's exactly the same Father Christmas we know in this world, even though we never see him in person here:

And on the sledge sat a person whom everyone knew the moment they set eyes on him. He was a huge man in a bright red robe (bright as holly-berries) with a hood that had fur inside it and a great white beard that fell like a foamy waterfall over his chest. Everyone knew him because, though you see people of his sort only in Narnia, you see pictures of them and hear them talked about even in our world — the world on this side of the wardrobe door. But when you really see them in Narnia it is rather different. Some of the pictures of Father Christmas in our world make him look only funny and jolly. But now that the children actually stood looking at him they didn't find it quite like that. He was so big, and so glad, and so real, that they all became very still. They felt very glad, but also solemn. (LWW, "The Spell Begins to Break", pp. 98-99 in the Puffin edition)

As @coracle says, the name Pevensie isn't given in this book — it's first used in VDT, if I recall rightly. But Father Christmas addresses Peter, Susan and Lucy by their first names without hesitation, and so does Aslan when he meets them two chapters later. I didn't actually remember that in the film, Mr Beaver prompts Aslan, which really doesn't make sense and is yet another silly slip that I hope Netflix won't repeat when they get to it!  

Looking through my movie companion book(s), LP is on Lucy's cordial and PP appears in the design on Peter's shield. 

@mel, That's cool as well! I never noticed! I guess it would be weird if only Susan got her equipment initialed....