Ok I frequent religious/circles and the concept of a twin flame is basically this with certain variations:
Your twin flame and you are basically the same soul split into two, who then incarnate lifetime after lifetime until eventually once some arbitrary point of spiritual development is reached you come into “Union” is reached and then spiritually ascend together (in some interpretations this will be your last incarnation).
The mythological root of this whole idea goes back to Plato, who talked of the Gods being frightened of the potential of mankind and split the original two headed/ four limbed human into man/woman. We then spent our lifetimes frightened, bereft and alone until the literal split halves find each other, experience a strong pull back towards one another and an overwhelming sense of home and completeness.
Various other new age ideas have and can be tacked into this paradigm. Lots of people who feel they are on the twin flame path feel they and their twins have special missions that are supposed raise the collective awareness and development of humanity.
There is an entire cottage industry sprung up around this concept of twin flames amongst tarot readers, life coach types etc etc. However all too often it seems to be a lot of lonely and heartbroken people, waiting for that destined person (called the separation phase), or worse case scenarios putting up with abusers because somehow their unconditional love will win the day in the end.
One area I can get behind though is apparently you won’t attract your twin flame into Union until you can truly love yourself healthily. So anyone who chooses this as their paradigm who are on a road to self acceptance, self knowledge and self actualisation. Who am I to argue?
Hope that helps.