Battery save function does nothing besides activate dark theme, reduce background activity and delay notification.
Meaning – Nothing.
Absolutely Nothing to keep battery last longer.
Nothing is the name of the phone so OK with that.
What would make the battery last longer is something completely different.
That would be a limiter to stop battery from overcharging just like electric cars have.
Normal charge would be stopping charging somewhere between 80 and 85% and that would keep the Li-Ion battery healthy many more years than if constantly charging to 100%.
That could be called “Smart Charging”.
Samsung (and probably more companies) know that, and it is something that can be activated. Not something done automatic, so those that want a 100% charge can have that.