I am getting an error message "Something went wrong while saving" when I am trying to save. I can use save load option, but if I close the whole game I will lose all my save data and I can only start new game.

Any idea how to fix this?

It only happened to me once, at the start of the final mission. I play on Linux. The savegame was completely broken and loading yielded a black screen (although the interface was visible and characters responded to keypresses). I loaded a previous savegame.

Technically the game doesn't really have a concept of "playthrough". Missions are getting unlocked and you can select any of the unlocked from the main menu. No skill points to assign or other forms of meta progression. You'd need a major data corruption to lose ALL your progress, and if that's the case it's worth checking for hardware defects.