I think you might be approaching this idea too much from a western, heterosexual mindset.
If this were something that has always existed in a society, two things can be said: one, it probably arose for an evolutionary reason, and two, it predated the establishment of culture.
There are many species in which gender works differently. I am far from a biologist, but I am aware many aquatic life forms and insects either have unusual versions of the common male/female, are hermaphrodites, or can change gender on command. The main reason, to my understanding, that this does not occur in larger life forms is that they posses larger, more complex reproductive systems, making changes more difficult, and possessing both sexual traits more costly to physical resources.
However, if a life form was developed with uncontrollable gender changes, its culture would develop around that. Within anthropology, it is often said that gender is a social construct. Many societies possess third genders or allow switching between them. Obviously, this is a cultural construct, and does not involve physical change in sex (although these days, it can mean that). While it is more politically contentious, based on that I would say it stands to reason sexuality is also a social construct.
What that means for your question is that the idea of being a given gender in the first place would likely never arise if sex was not fixed, or if it did exist, would be based on some other phenomena (perhaps choice). Someone might identify as "female" despite actually being male half of the time. Similarly, it seems probable everyone would end up bisexual, due to physical necessity. As such, there would be no particular reason to believe marriage would work any differently on that premise alone.
However, it may indeed impact the idea of marriage. It could lead to the idea never developing, for one. It could also lead to three-party marriages, or some other system. It is almost impossible for me to predict the impact it would have on marriage, though, because marriage is already defined very differently in some parts of the world. Throwing something as large as changing genders into the mix would confuse the situation beyond what I think a single, clear answer exists for.
Of course, I just assumed this was always the case. It is also possible that those possessing this gender changing effect were created out of an already existing species, like humans. In that case, I would suspect that at least for some period of time, our cultural constructs would persist - polygamy or simply abolishing of marriage seems the more likely short-term result, as sexuality and gender are more deeply rooted in the cultural mindset.
To address the matter of changing how people look; really, there is no reason to think people would look outwardly any different, let alone different enough it would be an issue. Especially if you're posing this as a genetic phenomena, there would not be a major physical change over that short a time, if there was even any at all. Perhaps if it was magical, and there was a total change in body structure, but even then I can't imagine it being more confusing than someone dying their hair. There would still be the same dress, mannerisms, etc..