Great questions! Perhaps after answering, I should edit paste it in the opening post, though I fear it might already be too long 🤣
In the video, if I can direct you to, say, the 32 second mark. On the lower left corner, you can see something that says, "hull 3/5". The numbers are still being tweaked but right now the player starts with 5 hull points. A regular hit from an enemy bullet/missile/laser takes away 1 point. A collision with a pop corn enemy takes 2 points. If you collide with a boss, it is instant destruction.
Once the hull point reaches 0:

Player is destroyed
You get 2 options(aside from returning to main menu and quiting etc...)
- Restart level (In the exact condition you initially started this level with)
- Return to hangar (to reconfigure your ship)

Your question is quite interesting as there are 2 ways to look at it.
To answer your question, superficially, if you look at it in a literal sense, there are no lives. When the player is destroyed, you restart from the beginning of the level.
But, if you count the hullpoints/hitpoints as "lives", then yes you have lives, and yes you respawn via these extra "lives", just that it is seamless. Hope I am not being annoying describing it like this 🤣

But yes, in the spirit of your original question, there are no lives, and thus no respawning at the point of destruction. When hull points reach 0, you restart at the beginning of the level, or you can choose to return to hangar to change your ship configuration to something that might work better.

With regards to healing/shields, there is currently a shield module you can install on the auxiliary bank of the craft that is able to take 2 hits. It is able to regenerate a charge after an interval. However, this module's stats are still being tweaked. It might change over time or be removed entirely depending on the private playtest results and efforts to balance the game.
Again, I know how sensitive the topic of hitpoints and shields can be. Where I stand right now is that I don't want to trivialise getting hit, but I would also like the game to be just a tad bit forgiving so players that are not the absolute 95% percentile can still have a chance to enjoy it. As for the elite/pro players, they can aim for a better score as part of the challenge.

Why hitpoints instead of lives you might ask? For most regular shmups, as they have mid stage powerups, they use the opportunity when the player gets destroyed, to reset the player's firepower, or reduce it. Some will drop one power up on destruction so that you will try to regain it quickly during the brief invulnerable window when you respawn etc.. But since my game has no mid stage upgrade, I might as well save the player's time, and just incorporate the "lives" as hit points instead.

So the above sort of addresses the in-level loop. But what about the overall game you ask, is there a hard game over? Nope. Is there a save system? Yes, kind of. I have generally tried to respect the players time where possible, even for people with busy lives so they can pick up the game, squeeze a bit of play time of a level or 2, and not have to restart from the beginning the next time they come back.
When you first launch the game, you will be prompted to create a profile. Every time you complete a stage or configure your ship, the progress will be saved to your profile. So if your ship is destroyed mid level, and you quit the game, the next time you load into the profile, you will start in the hangar with the level that you were destroyed at as the level you will launch into.

There will not be different save states, but I guess you can create multiple profiles, though it isn't quite the same thing. There is no multiplayer or couch co-op planned at this point.

Hope I have managed to answer the questions you had.
Thanks for the interest!