If you were a infertile woman would you feel like not a real person?

No. Never. That attitude presupposes that procreation is the purpose of the human race and that something else is less... I don't believe that for a minute. Not having children means you can focus on other positive attributes of yourself and contribute to the well being of others in a different way.
Very likely. Fetility is such a part of human nature that when there is dysfunction, both parties can feel inadequate, loss of gender identity and many other emotional symptoms. Many are depressed, angry, or withdraw. These are normal feelings but should be discussed with family partners or a listening physician.