Is the urge to awaken the same as the question "Who am I?"
I think one and the same in the sense that it is the urge itself truly asking that question. So there's a feeling and knowing that can come from that urge, not from who I think I am.
And how to be aware of that question emerging from such an inner depth?
Maybe a special sensation, a knowing that a very special process is in play. Maybe that's what spiritual maturity amounts to, a distinct sense that something of the greatest value is getting done regardless of our awareness of it.
If there's no awareness of it, what's the point of it; how are we actually involved?
Awareness yes; perhaps not continuous by far, yet momentous at the least. Also this simply underlines an acknowledgement of the extent of our self-delusion, of ego-mind's dominance.
And you would call that spiritual maturity?
Yes, in part. As this attitude grows, the doubt that anything can truly change decreases, opening the heart to feelings of gratitude, trust and a deeper knowing.