I have a hole in the hardwood floor (I think it's birch) and I'm thinking to fill it with wood filler.
Is that a good idea?
It's probably 2/3 mm in diameter.
It looks like a knothole that gave way.
Filling is complicated by the color difference between the body of the wood and the dark ring of the hole. You could put in a dark filler, as if it were the knot. Or you could scrape away the darker edges and put in a lighter filler similar to the color of the plank.
Either way, you would need enough filler to make sure the bottom of the plug is sitting on the subfloor. The adhesion of the filler to the sides of the hole would not be enough to keep the plug in place. Even with this, the plug may shift or break away.
An alternative is to cut out a slightly larger hole and insert a dutchman, a new piece of wood use to fill a broken or missing section. In fixes like this, the inserted piece would usually be a pointed oval (sometimes called a marquis shape). It also should have a slightly tapered edge with the top being wider than the bottom. The corresponding hole in the wood has a top surface wider than the deeper section. This gives more glue area and add structural support to prevent punch through. All of this assumes you have access to some of the original wood from the floor or a comparable piece.
Obviously the patch system is a bit more complex than the filler and requires a careful sanding and refinishing of the immediate area.