Does general anesthesia in surgery makes you really asleep?

No: General anesthesia is not the same as regular sleep. It is more a state of controlled intoxication where you are not conscious of what is happening to you. General anesthesia is much safer now than it was even 30 years ago before pulse oximetry became widely adopted. Between better drugs and better patient monitoring it is very very safe.

General anesthesia: Your question is not clear. Do you mean asleep during surgery or after surgery? It can let you out completely, so you will not feel anything during surgery.

Yes: This is not your sleep as in night time sleeping, but a deep sleep in which the surgeon is able to operate without you noticing.

Yes and more so: Its beyond sleep. More like a comatose state and the drugs/meds used in anesthesia affect the brain waves differently and the wave patterns will look differently than someone just sleeping and this makes sense because the requirements for the surgery are different than having a sound sleep.