"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Words such as courting, going steady, seeing someone would be widely understood, but may seem outdated or incongruous to some people.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.)
Deriving from your own explanation in the OP, the natural choice would be amicular.
I do not seem to find any dictionary entries. Need to see why.
Preliminary:
Book Doctor Gwen : 92 Feminine and Masculine Word Pairs
Feminine term / Masculine term /// neutral or inclusive term
4. amicular* / avuncular
(*Terms that are slang or recently coined.)
Contemporary Pragmatism - Google Books Result
books.google.com/books?isbn=9042018445...
John R. Shook, Paulo Ghiraldelli - 2004 - Philosophy - 200 pages
... be offered as amicular advice to discourse generation researchers, along the lines of the earlier 'Don't ask for the meaning; ask for the use', ...
Best Answer
The word you're looking for is enate.
From The Free Dictionary:
e·nate (ĭ-nāt′, ē′nāt′) adj. 1. Growing outward. 2. also e·nat·ic (ĭ-năt′ĭk) Related on the mother's side. n. A relative on one's mother's side.