I have a friend, a widow, who had many years of a good marriage. She has done a remarkable job building a new life. This year she went skiing for the first time in decades. My reaction was, "I am so proud of her."
One of the definitions of pride is a reasonable or justifiable self-respect.
I shouldn't be proud of her. I had nothing to do with her courage in living. I wasn't there during her times of self-doubt and often not there when she pushed herself to do something new or difficult.
When my daughter has some new accomplishment, I may feel a sense of pride on the off chance I did something right in raising her, but deep down I know, she accomplished the things on her own. My part may have been placing things that helped her should she decide to use them, like a good education, but I never conjugated her Latin verbs or passed the accounting exam or wrote her masters thesis. She did that.
Then I came up with a different phrase...I am so proud for you...in the sense that I recognize the accomplishment and want you to know I care and celebrate what you've accomplished.
Hope that is a better phrase.