Although it can be difficult to put an exact origin to the idea of romance, the origins of romantic love are somewhat more clear. Romantic love came into being during the Middle Ages around the Mediterranean region, where the concept of courtly love began emerging in the Arabic and Latin cultures. The concept of romantic love later started gaining popularity across Europe during the Renaissance, around the time of Shakespeare. Previously, love was generally much less spontaneous and much more structured. Couplings were decided upon by elder family members for the good of the family as a whole, rather than the two individuals primarily involved. It's important, however, to distinguish between the origins of "love" and those of "romantic love". Love can be seen far earlier than romance, one need only look to Odysseus' quest in the Odyssey for proof of his devotion to his wife, and similar tales exist in other ancient epics such as the Ramayana. For our purposes, what's important is not that he loved, but that he did no "romancing" to secure that love.
The major distinction between the time before romance and the time thereafter boils down to choice. To what degree is one person able to choose a partner? This view is suitable to some degree, though it has some obvious flaws. Take Harvest Moon 64 as an example. In this game the protagonist is able to choose over several years which of the local women he will court. To this extent, the choice definition fits quite well, as it does for other games like Knights of the Old Republic. The definition falls apart, however, if we look at a game with fewer choices. A story with a structure that limits the number of suitable partners to one can still contain a great deal of romance. King Kong makes an excellent example of this motif. In this story, the woman, whose name varies by the retelling, is taken by the massive ape. Invariably, however, the woman grows fond of the ape such that by the story's end, the death of Kong is a tragedy. Here again, we see definite romance despite a clear lack of choice on the part of one party. This applies to many other games too where the circumstances more than anything else dictate the number of available partners (your Final Fantasies, drop ship pilots, etc.).
By the nature of the extreme circumstances of many video games, much of the romance they portray seems fated, rather than elected by the participants. Fortunately for such romances, though, video games rarely follow them beyond the circumstances of their beginnings.