germany
There has been quite a journey, since this form has been used to try to enable unmarried same-sex couples to get the same rights as married heterosexual ones.
However, same sex marriage is legal for a while now. So the courts have focused in their definition of "marriage like relationship" on the status and privileges (and duties) that married couples have under the law and tried to grant similar rights (and especially duties) to those not formally married. You might cynically say that the state is mainly interested in making money, so there are a lot of duties, but not a lot of privileges gained from being a non-married couple.
So without all the history, let's jump straight to 2006, the last time this was changed/clarified. Keep in mind this court decided on duties, not privileges, of non-married couples:
Eine solche Einstehensgemeinschaft liegt nach § 7 Abs. 3 und 3a SGB II vor, wenn
eine Person mit dem erwerbsfähigen Hilfebedürftigen in einem gemeinsamen Haushalt so zusammenlebt, dass nach verständiger Würdigung der wechselseitige Wille anzunehmen ist, Verantwortung füreinander zu tragen und füreinander einzustehen.
Ein wechselseitiger Wille, Verantwortung füreinander zu tragen und füreinander einzustehen, wird vermutet, wenn Partner
- länger als ein Jahr zusammenleben,
- mit einem gemeinsamen Kind zusammenleben,
- Kinder oder Angehörige im Haushalt versorgen oder
- befugt sind, über Einkommen oder Vermögen des anderen zu verfügen.
Translated:
According to § 7 (3) and (3a) SGB II, such a "community of responsibility" exists if
a person lives in a common household with the person in need of assistance who is capable of working in such a way that, according to a reasonable assessment, the mutual will to bear responsibility for each other and to answer for each other can be assumed.
A mutual will to bear responsibility for each other and to stand up for each other is presumed if partners
- live together for more than one year
- live together with a common child,
- care for children or dependents in the household, or
- are authorized to use the other's income or assets.
There is no mention of sex at all, or of any gender or even the fact that it should be exactly two people. This is about social and financial dependence, and about the fact that married couples (who have to depend on the other if one of them still has income, instead of receiving social benefits) should not be worse off than non-married partnerships.
Please also note that courts have ruled that just living together, without any financial entanglements, for example sharing an apartment where each party pays their rent and their food, and they have contracts on how they divide up payments for internet or water bills, is not a marriage-like relationship. It needs to have this factor of shared responsibility over something.
A whole different topic is to prove that one lived in a marriage-like relationship for family immigration purposes when they were not allowed to marry in their country of origin, for example gay couples in countries where gay relationships are illegal and marriage would be evidence of a crime. For those laws and regulations of what constitutes a "marriage like-relationship", we would need at least a country of origin. I know it's customary to write answers for all countries, but that would be answers for all countries multiplied by all countries. Immigration law might also change independently of German social/financial definitions. That is just too much. For that kind of information, the Expatriates SE with exact information about a specific country and situation might be more appropriate.