I am like you too! I'll be 100 percent alone in the shower or in my car and I just won't be able to cry. The only time I cry is when I've been drinking. Our families sound very similar as well. I often got "Stop the poor you act" or "Grow Up" when I cried in front of my parents even at a young age.
The only time I'm able to let go and cry is when I've been drinking. Then I'll cry rather hysterically and even in front of people.
I think your therapist is accurate in her assumption that you stuff your feelings down or feel some shame in displaying raw emotion. I'd say it is pretty deeply embedded within you as you have a difficult displaying emotion even when alone.
Perhaps you could take up writing? That way you have a way of getting your feelings out in some way...and the next time you feel the need to cry, just let yourself cry. There is nothing wrong or shameful about it. And sometimes it really can make you feel better.