The same thing happened to us. My wife wanted to refill the dispenser, from the Bottom, and when she twisted and pulled, the bottle broke off, but the threading stayed inside the hole of the sink. Its stuck inside, and I've tried everything. I can't seem to get it out... and further tips would be very helpful
Thanks!

Something to try.

Remove pump mechanism. Assuming that the soap dispenser body is brass. From underneath the sink, heat soap dispenser body with a hair dryer. With small, flat bladed screwdriver, pry between the broken plastic bottle neck and the body of the dispenser. The heat will soften the plastic.

If you have an assistant, the assistant can pry with the screwdriver from the top and push down while you heat from underneath. Wives do not usually make good assistants in task like these.

You may or may not be able to find a replacement bottle. They seem to be brand specific. You may have to replace the entire soap dispenser.

If the nut underneath the sink that holds the dispenser in place is a wing nut you may be able to remove it with your hand. If it is not, you will need a set of tub faucet wrenches.

Soap dispensers are filled from the top. You just remove the pump and pour the soap in from the top.

Thanks to previous posters for the answer! Heating from the top with a hair dryer took about 3 minutes to soften the soap and plastic, and the stuck screw top popped right out.

I guess I just don't understand. You said my suggestion worked for you but yet you rated my post inaccurate. Just don't understand.

Apologies, my friend. No idea how it received a mark "inaccurate". Gremlins at work on my mousepad? It was an excellent answer !