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.
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.