Religious identity: who decides?

Sightings, a University of Chicago/Martin Marty Center publication, writes of the thorny dilemma currently being lived through in England around admissions to Jewish schools. Who gets to decide, when the competition is steep, who is Jewish? Is the test for that decision a practice-oriented test or a test of belief or a test of matrilineal heritage? And what happens when a country’s legal system is entangled in the decision-making? I do not know anything about English jurisprudence, but it strikes me that this question becomes most pressing when tangible, material benefits authorized (and in some cases provided) by the state are involved. I actually think that the current debates raging in the US around marriage for same sex partners have a lot of analogous dilemmas, since marriage is a state? a practice? that civil law uses to authorize certain benefits to some people.