You ask:
what playable character races are monogamous, polyamorous, or asexual, and how does that mindset affect the campaign
This is a matter of world building and character building.
As a DM
As a DM, you are free to do as much or as little worldbuilding on the subject as you like.
You can spend time detailing these subjects for your world, or you can completely ignore them. The mindset of reproduction only affects your world to the extent that you want it to.
The rules
Within the 5e material, there are clues for most races as to what is canon. For example, Volo's Guide to Monsters offers some detail on kobolds:
Kobolds choose mates primarily for convenience. Their lack of emotional bonding means they have no concept of marriage or permanent family relationships.
You decide to what extent that matters in your world. There are clues to many of the other races, and you can choose to fill in details as you wish. For instance, in the description of dragonborn in the Player's Handbook, apparently they lay eggs. Are they mammalian egg-layers? It doesn't say. You decide, it's your world. Perhaps in your world they give live birth. For any race, you can choose to accept the canon, change it, or ignore it.
Sometimes GMs create some documentation such as a campaign guide or a world guide or a character creation guide where they spell out some of these things for the benefit of their players. My own recommendation is to keep such things reasonably lightweight, and only detail what you feel you need to, to help your players create their characters and understand your world. Some players are likely not going to be very interested in a lot of world lore that doesn't directly affect their characters or actual play.
Player characters
In general, the DM designs the stage, the players define the actors. The Player's Handbook, in "Personality and Background", under "Sex", says, in part:
You can play a male or female character without gaining any special benefits or hindrances. Think about how your character does or does not conform to the broader culture’s expectations of sex, gender, and sexual behavior.
Obviously, you want the game to be fun for your players. Again using kobolds as an example, if you have a player who wants to play a kobold, you are free to define kobolds in your world, but you should allow the player to define the outlook of that character, and differ or conform to your worldbuilding as they wish. You define the species, but they define the character. For that matter, what if you think kobolds lay eggs, but your player wants to have been born live? You might consider how to make that work. Maybe this particular kobold was born to human or dwarven parents. Magic, right? In real world literature, stranger things happen all the time. There's a kid's book, Stuart Little, where a mouse was born into a human family. How did that happen, am I right?
I ran a campaign once where one the the PCs was a delightful and memorable female kobold cleric, with a moderately developed backstory. It was a great campaign and we had a lot of fun, and issues of kobold reproduction just didn't really come into the story, really at all, that I remember. I do remember when the cloaker dropped onto the kobold and the player triumphantly cast harm on it, killed the cloaker, pulled the becloaked and blinded wizard back from the cliff edge, and saved the day. It was a triumphant moment and one that we remembered and talked about. The kobold's outlook on romance and egg-laying? Not so much.
You can even go a step further and allow your players to help you define these issues. I mentioned dragonborn earlier. I was in a campaign once where one of the characters was playing a female dragonborn. A conversation something like this took place:
Player: Look, I can't tell if 5e dragonborn have breasts, or not, in past editions they did, in 5e it seems like WotC has tried to make dragonborn breasts less prominent in artwork, or maybe non-existent. On the internet there's wars. Reptiles! Gender stereotypes! I just want to know.
DM: I. Hmm. I don't think it matters to me. What do you think? Do you want breasts?
Player: I think dragonborn should lay eggs in clutches and nurse live, so I think they should have breasts. I mean, it's not a big deal, but that's what I think. I'm cool with however you want it to work.
DM: Okay. You just described how dragonborn work in the city you're from. Since that city is prominent in the campaign, that's how dragonborn work around here. There may be dragonborn somewhere else that are different. I mean, dragonborn, right? There's a lot of different lore about how they were created.
Player: (satisfied) Great, thanks.
The player went on to develop a bit of backstory involving the character's family and was very satisfied. Dragonborn reproduction was really a very peripheral issue, though, and the game moved on.
As a player
What matters is the character you're playing in the game you are playing.
You are free to decide your character's reproductive outlook. If this is important to you, work with your DM to clarify the details.
I can't stress this enough. You can read all the books, rules, and lore you want, and it can be very helpful and interesting, but what matters is the game you are playing in.
You may find your GM has very specific opinions, or very loose ones. Work with them. They surely want everyone to have fun, so they are likely to be pretty accommodating.