Tile transition strip damaged

I would break the rest of it off and mill a thin piece of oak with tapers on both sides. Then I would stain it to match the floor and install it so it overlaps the tile a little. If the floors are a bit uneven, rabbit it so it fits flat.
You might find a ready made transition strip in the flooring aisle at a big box.
That piece likely has an inch wide leg under the tile.

If I had to guess, I'd say the wood that's running the other direction from the rest is the transition strip, but it's not installed right from the looks of it. The wood bit should go up and over the tile edge, it shouldn't be a hard 90* like that (or at least that's how it looks in the picture.)

If it were my floor I'd try to pull up the strip of wood at the nearest "edge" (against a wall/door frame usually) and take it in to find a reducer transition piece that matches the color. Looks like a light oak - should be easy enough to find a match - you just have to find one that's a bit wider to go over the tiles.

The transition strip is likely set in the mortar under the tile. You can search "schluter floor profiles" if you want to see examples of how it is commonly done. Except looks like they cheaped out and used plastic.

An oscillating saw will make quick work of removing what is left over.

Is the tile higher than the hardwood ?

Home Depot or Lowes will have wood transition strips. Your hardwood looks like oak, so that is very common and easy to match. Might have to modify the transition strip.

The hardwood moves with changes in humidity. Caulk would be a better choice than grout.