That stuff is a serious dye.
Maybe check with wood working persons too.
Disclaimer - try at your own risk - do not know what this would do to any paint or wood or finish exactly.
I would think you may be able to remove excess color/dye with rubbing alcohol with q-tips or use denatured alcohol which will damage finish so only maybe try it out on a spare piece of floor board or other chunk of cabinet. Hairspray gets out pen but this probably soaked deeply into the grain.
Some salvage stores like ReStore have furniture pieces they break down & throw out or buy something cheap, CL
freebie, etc. You can perhaps replace the door with a brand new or "new" one from a used/thrift/salvage place. Or exchange it with one from another place in the house.
Best bet would be expert repair of floor and replace of door.
Floor can be expertly repaired as well by removing the damaged portion.
You may be able to you tube it if you are pretty handy and replacing the part that is stained with a whole floor board or a part of the board(s).
>>>>>>>I would be afraid all the cleaning that you tried would just increase the mass of blue to everything around it.
Dreadful thing to have happen. How angering and frustrating.