Jack, a citizen of Halloween Town, grown and raised there, has Halloween Colored Glasses. His POV in life revolves around Halloween. His thought process is spooky and scary and everything that goes bump in the night. The song in which your screen capture comes from revolves around Jack's Culture Shock, trying to figure out What's This? in regards to the very non-spooky Christmas Town.
You tell someone that has never heard of Christmas, who's mind is set on Halloween, that a magical person called Santa Claus exists, what is the first thought that they will fix on is Claus, which is a homonym for Claws. "When you're a hammer, everything looks like nails".
That said, the video above, when Jack first sees the Shadow, is at a distance, and the Shadow bounces a little, and the fingers definitely elongate to a claw like shape.
