Its complete rubbish. Any modern anesthesia will at the very least be accompanied by monitoring the heart rate and you can see the respirations. They would both spike to crazy levels early on and scream to the medical personnel something was wrong.
This is an old urban legend that has been around forever sometimes accompanied by conspiracy theories that people who go through surgery experience everything but have amnesia from the drugs so don't remember.
Not to mention if you could feel someone cutting a baby out of you or removing your spleen shock would set in almost immediately. This is an old and tired urban legend that is just silly and propagated by people who don't understand how anesthesia or the human body works.
Like all urban legends people will defend them as true to a crazy degree because they know someone it happened to. They don't.