Why do I feel like suffocating when I get filling at dentist?

Panic attack? Anxiety at the dentist is quite common. There are many things in your mouth at the same time. You must give up control. This is not unusual. Add protective barriers, being surrounded by people. Add some claustrophobia and that your ability to swallow is being inhibited, that you are numb, and that perhaps you have trouble nose breathing. You can sort this out one step at a time.
Anxiety. It's normal to feel short of breath during periods of stress and anxiety. Moreover, there's an element of claustrophobia when the dentist is hovering over you, inches away, with a drill and other scary looking equipment. Finally, you may be unconsciously holding your breath when he/she's working in your mouth - so there are many reasons.