No, but it does make the result identical. Unconscious.
But the condition isn't the only result.
The results of falling asleep:
1. You have the Unconscious condition, which ends when you awaken (DM's discretion).
2. You may possibly fulfill your sleep requirement (e.g. 8 hours/day).
The results of falling unconscious after dropping to 0 hit points:
1. You have the Unconscious condition, which ends only when you regain hit points.
2. You make a death saving throw at the start of your turn unless you are stable.
3. Any damage you take counts as a failed death saving throw.
4. Any damage you take from a critical hit counts as two failed death saving throws.
5. Any damage you take that is greater than or equal to your maximum hit points kills you instantly.
Identical results? Not really.
Yep. The attack creates an extra way to wake you up, and that's by healing. That's all it does by RAW. It doesn't remove a loud noise from waking you up. It doesn't remove another attack from waking you up, though damage at 0 hit points has other rules attached.
You think that because sleep imposes the Unconscious condition that anything that ends sleep also ends the Unconscious condition, even in situations where the condition was imposed by something other than sleep. This is not sound reasoning.
Unconsciousness imposed by sleep ends when the sleep ends. Sleep is ended by things that end sleep.
Unconsciousness imposed by dropping to 0 hit points without dying ends when you're no longer at 0 hit points. You're no longer at 0 hit points when you regain some hit points.
Initiative isn't tied to either unconscious condition. That's your house rule.
No. My house rule is that sleeping creatures are wakened by the commotion caused whenever hostile action (rather than hostile words, thoughts, or intent) is taken. Fictionally, this commotion takes place
before initiative and is the event to which the initiative check measures reaction. Before initiative is ever rolled, the creature is wide awake due to some commotion or other. This is my house rule.
The chain of events is as follows: At the table, a player (or the DM) declares hostile action for his or her character or monster. >> In the fiction, the character or monster begins to perform said hostile action, causing a commotion considered equivalent to a loud noise. Anyone within audible range is now alert, and if they were sleeping, they are now awake. If the character or monster beginning to perform the hostile action is hidden, anyone unaware of its presence is now surprised. This
fictional event is the Start of Combat. >> Still in the fiction, alerted creatures react to the Start of Combat, beginning to perform any actions they take in response. >> At the table, the DM calls for a DEX check, called Initiative, to determine whose competing actions are resolved first.
Right. The only added way to wake up someone who is unconscious comes from the combat rules, and that's healing to 1+ hit points. So sleep has X ways to wake someone up, and 0 hit points has X+1 ways to wake someone up. At least as written in the book.
As written in the rulebooks, there is exactly one way to awaken from the unconsciousness caused by dropping to 0 hit points without death, and that's by regaining any hit points, even one. The unconsciousness caused by falling asleep, on the other hand, can be ended by anything the DM has determined. It would be a misreading of the rules for the DM to decide that anything that awakens you from sleep also awakens you from the unconsciousness caused by dropping to 0 hit points without death, although I could see allowing a character to momentarily regain consciousness to utter some last words, for example.
Oddly enough, I just noticed in looking this stuff up that if you are out of hit dice and don't have magic healing available, you can't ever wake up naturally from 0 hit points. You regain hit dice via a long rest, as well as your hit points, but the long rest rules specify that you must have 1 hit point to gain any benefits from a long rest.
This has been addressed up thread. The creature with 0 hit points must be stabilized. Then, after 1d4 hours of remaining stable, the creature regains 1 hit point.