"Relationship" is a term with multiple meanings.
Marriage itself is a relationship. So when you are asked "Are you in a relationship?", an appropriate answer would be "Yes, I'm married".
There are other relationships, such as brother and sister, parent and child, colleagues, friends, tenant and landlord, and so on. However "are you in a relationship" usually refers to a monogamous relationship with a boyfriend/girlfriend -- you wouldn't answer the question "Yes, I have a sister".
There is no commonly-used word that you can just plug into a sentence that means "in a monogamous loving relationship but not married".
If the couple are living together, you might say they are cohabiting, more informally shacked up, or simply living together.
Words such as courting, going steady, seeing someone would be widely understood, but may seem outdated or incongruous to some people. ("Courting" is the language of my grandparents' generation; "Going steady" is the language of a 1980s American high school sitcom, etc.)