Does the government have the right to decide what the conditions are for what it will and will not "recognize" as a marriage? Yes or no?
Does the government the right to draw that line of distinction or not?
[Correction] Short answer is Yes, however I understand you trying to lay limits on the question, but your attempt to add absolute limits are artificial and not a function of the law (nor of common sense).
For example:
#1 If the conditions are implemented in a Constitutional manner, applied equally - Yes. For example, States set age limits that are applied to all applicants. As another example many states required blood tests as part of obtaining a Marriage License (IIRC Mississippi was the last to repeal it 6 or 7 years ago). They are/were Constitutional because the laws were generally applicable and not written to exclude certain persons or segments of the population.
#2 If the conditions are implemented in an Unconstitutional manner, applied in a discriminatory manner - No. For example, States limited Civil Marriage based on race composition of the couple, those were held to be unconstitutional. States also limited Civil Marriage based on the biological sex of the couple, those were also held to be unconstitutional. As a third example, the Federal government attempted to pick and choose which legally recognized Civil Marriages it would recognize for Federal purposes, that discriminatory action by a section of government was also found to be unconstitutional.
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In the cases of #2, it was a branch of the government the found action unconstitutional and so nullified the actions of other branches of government we those actions were tested against the Constitution.
So unconstitutional actions by one branch of government were ended by another branch of government - so in reality it was still the government that determined the boundaries that could be used.
Remember, "government" does not equal just Legislative Branch. Executive and Judicial are also branches of government.
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