Does running MacBook Air always connected to charger damage my battery ?

No.

All batteries are meant to be used.

All batteries and all computers inevitably degrade. Inevitably fail.

Use your MacBook Air for what you bought it for.

Do keep current of course, as macOS 10.15.5 and later has some charging-related updates, and further updates appear likely.

And if your MacBook Air battery does degrade—as all inevitably do—Apple offers a battery replacement service.

Official reading: About battery health management in Mac notebooks - Apple Support