"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Words such as courting, going steady, seeing someone would be widely understood, but may seem outdated or incongruous to some people.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.)
Best Answer
There's a specific term for the relationship between people whose children marry each other: co-parents-in-law.
(Wiktionary)
The Wiktionary entry additionally includes usage notes stating that in normal conversation the generic "in-laws" may be used, although this could be ambiguous.
To avoid ambiguity in everyday speech you can rely on context, or use phrases such as "my son-in-law's mother" or "my daughter-in-law's mother".
Wiktionary also lists coordinate terms such as:
Co-mothers-in-law specifically answers your question: