Well, they're wrong. This could be why countries like China take the piss out of The West over these issues. As tempogain said, using the word "right" is just a convention. If someone decides to beat you up and take your stuff, hollering that he's violating your human rights is unlikely to have any practical effect.
As I've said elsewhere, it would be more logical to frame these things in terms of responsibilities towards others. A person does not have a right to air, food and water, but society at large may not deprive him of those things (eg., by polluting the air, proscribing what he may grow on his land, or misappropriating the water supply). He does not have the right to education, but others may not prevent him from becoming educated.
Here's the problem: when you frame these things as rights, the implicit assumption is that it's the government's job to provide them (or, if not the government, a State surrogate). Thus Americans in some locations can be legally prevented from raising their own food in gardens or collecting rainwater from their roof (google it). Until recently it was illegal to homeschool your children.
Anyone is at liberty to hire a Priest of Scientology, print a marriage licence off the Internets, and hey presto, they've got themselves a wedding. British Common Law originally recognized any such union as a marriage. The bottom line is that if you tell other people you're married, you're married. It doesn't really matter what the government thinks.
But 'marriage' is not a relationship. Living together as husband and wife (or husband and husband) is a relationship. The government rubber stamp is a complete irrelevance. I agree that it's not my business if people want the government to pat them on the back and say, well done, here's your gold marriage certificate; I'm just saying it's a prime example of a First World Problem.
You've got your logic backwards. If property inheritance law says that a man may pass on property only to a wife who is duly registered with the State, it's the inheritance law needs fixing, not the marriage law. However, I don't think that's even the case; if a man wants to will his property to his spouse (male or female), there's no law that prevents him. There is no law preventing someone from holding a wedding.
Adoption is a whole different issue because you are talking about the rights of the adopted child, not the rights of the adult couple. Nobody, male, female, married or otherwise, has a "right" to adopt a kid like they're getting a puppy from the pound.
Look, all laws are basically arbitrary and marriage laws are no different. It is not legal, in Taiwan, to marry your sister, a 14-year-old, or a water buffalo. At various times and places, these things have been (or still are) perfectly legal or even encouraged. When they're discouraged, it's either for a very good practical reason (eg., inbreeding) or because some other law is incompatible. That's what's happening here, with the examples you mention.
Note for the hard-of-thinking: I'm not arguing that a homosexual marriage is the same as marrying a water buffalo.
um, no. I was saying the exact opposite: if one already has the rights under discussion, why would we need the government to ring-fence it?
Of course not. But what I think should be, and what is, are two very different things. The reality is that you have a "right" to not be arbitrarily imprisoned only because the government has better things to do. Some governments have trouble finding alternative amusement and therefore imprison people arbitrarily. It passes the time. Talking of rights here is utterly meaningless because the supposed right is so tenuous; it can be violated on a whim.
What you are all arguing for here, IMO, is assigning powers to the State that it should not have. You are asserting that a relationship is only valid if the government says that it is.