I just had a flu shot this week and I tell you, one plunge and done. It doesn't take long to get the shot. Literally a second. I would guess you got the full dose. That's typically regulated and it would have to be a very unethical doctor to not give you the full dose. The speed of how long the shot took doesn't factor though because it IS super fast to get it.
It's really hard to say exactly what happened, and whether you received enough to be effective. It won't hurt to get an additional vaccine - just wait a couple of weeks after the first administration before you do.