In Three Houses, Monica was a student rescued alongside Flayn in chapter 6. Hanneman and Rhea explained that she was a student who disappeared shortly before her graduation the previous year, and was thought to have run away. Monica rejoined the Black Eagles class and immediately attached herself to Edelgard, which along with her constant cheerfulness made quite a few people side-eye her. (She also called Edelgard "Edel." Take note of that.)
And they'd be right to: a few chapters later, Monica stabbed Jeralt and fled to the Sealed Forest...where she revealed her true identity. Kronya, one of Those Who Slither in the Dark. Kronya would meet her own end when Solon sacrificed her to trap Byleth in Zaharas.
So from this information, we're basically told that the real Monica had been dead for some time and Kronya..."borrowed" her body in order to stick close to Edelgard. Now, given what we learn about Edelgard in chapter 11, we can assume that Kronya was sticking close to her in order to keep track of their plans.
If you bought the DLC or watched a video of Balthus and Hapi's paralogue, you get more tidbits about Monica and House Ochs. Namely, that Monica's father stole Balthus's family Relic Vajra-Mushti in order to save Monica, who was being held hostage by the Slithers. Sadly, he turns into a beast and you're forced to put him down, and Edelgard mentions that House Ochs hasn't been in good shape since the true head of the household died some time ago.
Cut to June 2022, when Fire Emblem Warriors: Three Hopes comes out. In an alternate timeline where Byleth never became a professor, Jeritza takes their place for whichever classs the new avatar, Shez, chooses. Kostas dies, Jeritza leads the class to the bandits' stronghold...and Monica is chained up in a cell there. The real Monica, whom Kronya had "plans" for. And who calls Edelgard "Lady Edelgard" or "Your Majesty".
(Kronya did not do her homework.)
If you choose the Black Eagles, you quickly learn that the real Monica was indeed close to Edelgard, and that her real personality wasn't that different from Kroyna's bad acting job. Monica spills the tea about her disappearance, Solon is unmasked, and this leads to Edelgard jettisoning the Slithers from the Empire rather than keeping them in her arsenal.
It's at this point I realized Edelgard and the fake Monica were not plotting together, but that they chose Monica for a reason: she's Edelgard's friend, and the Slithers wanted to keep tabs on Edelgard, whom they forcibly implanted with the Crest of Flames to turn her into their ultimate weapon and puppet.
So yeah. Three Hopes, in making Monica a playable character, shed more light on the Kronya arc of Three Houses, and that's pretty neat.
Monica is in love with Edelgard. This is canon, irrefutable. This isn't something the writers chose to leave ambiguous or only hint at, Monica makes her feelings for Edelgard very clear in both their B support and in her FE Heroes Forging Bonds event.
Now, said love is...somewhat extreme, a bit obsessive. Not Tharja levels of obsessive*, mind you, but Monica has a flawless memory and keeps track of everything related to her time with Edelgard.
However, while some might find this offputting or gross, Edelgard only reacts with a raised eyebrow. The same way she reacts to her other friends and allies, because she's among a lot of quirky people and has quirks of her own. She also clearly appreciates and cares for Monica, as seen here:
Sadly, Three Hopes doesn't have any paired endings. Due to the nature of it as a spinoff/alternate universe, every route has an open ending with the war still yet to be finished after each Lord achieves their main goal. The closest you get is Shez giving their Merc Whistle to a character and getting a letter for them at the end of the game.
But my feeling is that if it did have paired endings, Monica and Edelgard would have one and it would be very sweet and in line with their characters.
As you saw above, the last screenshot is from Golden Wildfire, during the mission to assist Edelgard as she advances on the Monastery to fight Dimitri. Edelgard can handle herself pretty well, she's fast for an armor unit and hits like a truck and her defenses are pretty beefy even if she needs a heal now and then.
Monica, however, is a mage, and mages are squishy even in their final forms. So when the enemies attack Monica, Edelgard commands the army to save her and that becomes the mission. Yes, Monica is one of her generals, but her wording is basically "save her at any cost, she's important to me."
Earlier in the route, you see Edelgard and Monica chatting about the Empire's plans. Edelgard explicitly trusts Monica, sees her not just as a valuable advisor and general, but values her input and her help.
You also see this early in Azure Gleam, when they discuss their loss at Ailell and Petra being taken in by the Kingdom. Sadly, this is the last time we see them together because Azure Gleam...does not go well for the Empire. All I'll say on that front is fuck Thales. We don't get Monica's feelings or thoughts on what's been happening with Edelgard after the timeskip, but my guess is she's either being kept in the dark or she knows, but after her own brush with the Slithers she knows trying to do something won't end well for her or Edelgard.
But I hold firmly to the belief that Edelgard recovers after the events of Azure Gleam and Monica is by her side every step of the way.
That changed when Three Hopes happened and we finally met the real Monica. Right away it was obvious she adored Edelgard and was completely devoted to her, and I'm a sucker for devotion ships. Especially in this franchise! I can't pinpoint the exact moment it jumped onto my list of Best Edelgard Ships, but it did. I was prepared to love Monica and even more prepared to love her and Edelgard's supports, and the game did NOT disappoint.
Monica, to me, is the other side to the "for Lady Edelgard" coin. Hubert shows his love by killing Edelgard's enemies, sneaking around to gather intel on any threats they'll need to handle as a team, and Dark Spiking anyone who goes up against him in Enbarr on all the non-Crimson Flower routes. Monica shows her love with cheerful praise, baking cakes, making tea, arranging flowers, and saying she'd be a willing sacrifice if it meant Edelgard could fulfill her goals. No, it's not entirely healthy - in fact, as mentioned before, Edelgard feels guilty as hell over the alternate timeline where she'd have let Monica die because the cards didn't line up for her rescue.
Edelgard appreciates Monica. She's a little "uhhh" over her intense feelings, but Edelgard is just Like That in general at times. She appreciates her friends, but she tries so hard to be serious and is surrounded by colorful characters. She really does adore Monica, and that's something I like in my devotion ships - reciprocation. Edelgard in Three Hopes is less closed off and more willing to be open with her allies, and being able to save someone she could have lost plays a big role in that.
(An aside: Read Walt's fic. It's not Monigard but it's a good example of speculation, headcanon, and using what canon gave you to craft a long story about a character who survived due to AU intervention. And he turned out to be right about a few things!)
It's not clear whether they're actual childhood friends or if they met when they were older, but I tend to lean towards the former because I also have a weakness for childhood friendships and how romantic relationships can develop from them. I like to imagine they met young and became fast friends, and all the fun they had with each other before Arundel and Duke Aegir pulled their stunt and ruined Edelgard's family and life.
Another thing I like about them is the contrast in their likes and dislikes. Edelgard loves sweets and hates to swim, Monica prefers savory foods and loves to swim. And I can easily see Monica pretending to eat the sweets Edelgard puts out for them, or learning to bake cakes or deal with going into a bakery because she knows how much Edelgard loves them and she'll endure anything for Her Majesty. (And then Edelgard finds out and tells Monica to just be honest, they can have savory snacks if she doesn't like cookies!)
Finally? They just look good together.
(Fanart by Eeby on Pixiv.)
The red and white contrast of their hair, the way Monica's pink dress picks up both, Edelgard's outfit in Hopes being red and silver as opposed to her red and gold look in Houses, the way the Three Hopes model renders are smoother and softer and the colors are more vibrant overall.