My dog did her best to try to claw my friends door open.
I have a new piece of weather stripping but I need suggestions for filling the deep scratches in the wood. It seems regular spackle would chip easily since it would most likely get bumped in the doorway.

Is the Bondo stuff they sell at the box stores what I need?

Wood filler/putty

https://thecraftsmanblog.com/the-wood-filler-epoxy-test-year-1/

If its going to be painted, we used automotive bondo. I would maybe drill a few anchor holes in opposing directions for a bit more anchor but the smallest we filled with it was gaps in veneers for bent pine jambs.

Hopefully they have some paint left. Takes a few coats to cover the pink or whatever color you find locally.

I've used DAP plaster patching compound. Apply, sand smooth when dry, paint.

Sand it, use wood filler, sand again, primer, and then paint.  I had to take a bunch of trim off an interior door a couple of months ago to squeeze some furniture through.  It got scratched up a bit taking off the smaller pieces of trim and now it you can't even tell.

Skip the wood filler and do the Bondo.  I’ve used the shit since the late ‘70’s and was a body man for 11 years.  I never thought about using Bondo until 7 years ago we hired a painter to paint the kitchen cabinets. I was going to sand and prep to save money and time.  Asked him what wood filler he preferred me to use.  My jaw dropped when he said Bondo.  Been using it on projects I plan to paint ever since.  
If your hardener that comes with it is red and not blue, put a coat of stain blocking primer on it first.