In my personal experience, "maximum heart rate" is a bit of a wildcard but the Mayo Clinic describes it as:
...the upper limit of what your cardiovascular system can handle
during physical activity.
I've always understood that as "the maximum heart rate your body can safely achieve." It's not like you drop dead one BPM over your "max", and there's way too much variability in the human population to simply do the "220-[your age]".
Long before you get near your "max", you'll cross the lactate/anaerobic threshold: the point where you cease aerobic cellular respiration and switch to anaerobic. Even someone well trained in anaerobic threshold training can only maintain that level for a short time (minutes, typically).
There's a 2004 study that really gets into some of this as well if you're interested.
Specifically answering your question of: Is it possible to run at your maximum heart rate?
Yes. Indeed, it's probably the only way you will ever get to your "max". I personally saw my heartrate monitor blinking 199 (meaning it was higher than 199) when doing hill sprints on my road bike during my training years ago. It hurts to push that hard, and I would encourage you to only push into levels like that with a solid training plan and under the eye of a competent trainer.