In many states you can have common law marriage with no contract/license at all.
Common-law marriage is actually more complicated than that, it's not simply another form of regular marriage. As a legal concept it only applies in some situations. Common law marriage does not give the parties to it the full set of rights and obligations a regularly solemnized marriage does. And for what it's worth, it's recognized by fewer and fewer states as time goes on. Mostly because it
is legally rather a messy set of concepts - not the nice, (comparatively) neat legal package that regular marriage is.
ETA: And fwiw, even common-law marriage involves a contract, though one implied by law, rather than one expressly entered by the parties.
As far as I can tell the actual marriage license isn't used for a whole hell of a lot
That's like saying the results of your road test aren't used for a whole lot. And for that matter they aren't. Just to establish that little thing called your "legal fitness under existing law to receive a driver's license." Same thing with a marriage license.
People also get married internationally all the time and the government figures out how to recognize their marriage even though they don't have a state issued license.
As a matter of legal convenience and in the interest of friendly international relations, sure, but they don't really "figure out" anything. they simply treat a marriage considered legal in the jurisdiction in which it was performed as if it had been performed here (as long as the marriage doesn't otherwise contravene US/state law or public policy.) US/state laws do not recognize the marriages between a man and wives other than the first one, even if they're legal in Saudi Arabia, for example, or those involving children under US ages of consent, even if they're legal in Bangladesh. (And if the question of the existence or legality of the foreign marriage arises, it has to be proved with documents and all sorts of fancy attestations and official seals and stuff - no one accepts just the word of one or both parties that the marriage exists.)
Moreover, nor do US laws grant the parties to foreign marriages the rights/obligations they would have under the law of the jurisdiction in which the marriage was performed, they grant them the rights/obligations that exist under US laws.