Whenever someone suggests that board games could use better, more visible, representation of same-sex relationships, a response I often see is that board games don't depict relationships - a reminder that to a lot of people relationships are only visible if they're same-sex. In reality, however, there are a variety of ways in which different-sex relationships are represented in board games.
Stone Age, Agricola, Caverna, and Dungeon Lords all mechanically represent different-sex relationships in how you grow your workforce. That they play coy by calling it 'family growth,' 'wishing for children,' or 'the magic room' doesn't lessen this, any more than depicting childbirth as a stalk delivering a baby in a children's book does. The mechanics of the deed don't need to be present for the implications to be there.
Other games are less mechanical in their depictions of relationships, using flavour text or artwork. The Marriage visitors in Viticulture depict a bride and groom, and we inherit our vineyard from a mama and a papa. In the base game of Sentinels of the Multiverse, Legacy's depicted interacting with his biological daughter, while I cannot find any reference to Tachyon's wife in the base game, only the promotional online comic. This is perhaps the easiest to fix, just needing a little art or flavour text - a couple holding hands, his boyfriend, her wife - making it especially disappointing when games only that depict different-gender couples in this way fail entirely at depicting same-gender ones.
A lot of games are set in European historical settings before the legalization of homosexuality. And while that may make same-sex relationships less visible, you do see examples in the historical record such as Julie D'Aubigny AKA Mademoiselle Maupin, who rescued her lover from a convent via a little grave robbery and arson.
There are games which attempt to depict relationships in a gender-neutral manner, to mixed success. Various cards in Fog of Love have tiny heteronormative assumptions. How much these assumptions negatively impacts the experience of playing a same-sex couple seems to vary from player to player, and to its credit, there are some scenarios in the base game that make more sense when playing same-sex couples. The game permits queerness but doesn't always succeed at fully accommodating it. Similarly, you can pick the gender of your romantic partner with zero mechanical impact in Pursuit of Happiness. This approach, however, feels similar to playersexual NPCs in videogames - where NPCs sexuality exists only in so far as they are attracted to the player character, falling short of real representation. The work to get same-sex relationships into the game rests on the player when handled this way.
Other games leave if relationships are same or different-gender to chance. When used in campaign games, there's a chance no such representation may appear at all when relationships are handled this way - In Charterstone, for example, at a certain point in the campaign one character gets the opportunity to enter into a relationship with a specific NPC, playing out identically no matter what gender that character is. And while it's wonderful that I got to play a character who, in our campaign, was in a relationship with another woman, but the game only has a 50% chance that this is possible, and has no guarantee that the character who has the chance to be in a relationship will be a character in the campaign.
Board games have a long way to go here, but some designers and publishers are trying to make improvements. With several games, some of which I brought up here, it feels like an earnest effort is being made, even if those efforts sometimes fall short of where I'd like them to land. But sometimes it feels like a struggle to get the discussion even to the start space - an acknowledging that board games can and do depict any kind of relationship, including games not specifically about them.