You might be able to pull it out with a hook.
You can unbend things like a wire coat hanger, a very large paper clip or a similar object and then bend it into a long hook that fits through the bottle neck.
Try to push the whole hook through the middle opening of the plug, then let one end of the hook wrap around the outside of the plug. You should then be able to pull it through the bottle neck.
Another solution that is way more fiddly is to use a string to pull it out.
First you need to bind a weight to the end like a small screw nut. Let the string down into the bottle and put the weight through the opening in the plug. Then gentla shake the bottle around so the string catches the plug and the weight falls back out of the bottle neck. You can then pull on both ends of the string to pull the plug out.
Assuming a plastic bag doesn't count as a "special tool", you could try this approach.
In case the video disappears, the short version (pictures are just screen caps from the linked video):
- Twist the bag and insert it most of the way into the bottle, leaving the top facing out
- Turn the bottle upside down and gently shake it - the goal is to get the stopper close to the neck of the bottle, nestled inside the plastic bag
- Blow into the bag, inflating it around the plug
- Twist the bag around the plug and pull the bag out, bringing the plug with it
Looks like plug is polyethylene. Acetone dissolves polyethylene, so get a bottle of nail polish remover (acetone), pour it in and wait till it dissolves enough to let it flow out.