Does the saying "No two who love each other separate except due to a sin committed by one of them" mean that separation between those who love each other is only due to sins, or can it be due to a lack of understanding or incompatibility?

Lack of harmony or understanding does not sever affection for the sake of Allah. Rather, it may be limited to abandoning partnership in worldly matters while maintaining affection. The aforementioned Hadith refers to the severance of affection for the sake of Allah due to injustice, abandonment, and the neglect of a Muslim's rights. Its context and continuation indicate that the separation between those who love each other for the sake of Allah only occurs due to a sin committed by one of them. Sins strip away the blessings of obedience, and if your companion changes, know that it is due to a sin you have committed.