Before backstories are written, before dice are rolled, talk to your players. Make sure that you explain your DM style, the theme of the campaign (I mean: "this is basically a dungeon crawller, murder hobos welcome", "this is horror campaign, you should expect vivd descriptions of violence, gore and torture", "this campaingin is heavy RP, we fight stuff, but the main point is creating dramatic epic story for your heroes", "this campaign is a mystery camapaign, if you don't like riddles and political intrege maybe you should sit this one out."). In that layout your character expectations and backstory expectations.
Maybe you're fine with War and Peace, maybe you are fine no backstories, maybe you're giving them backstories, maybe the backstories are all lost as the players start without memories. The point is, let them know what to expect and what you expect up front. Misaligned expectations cause issues like this, and the more your on the same page, the more you're going to have an easy time finding buy in.
Backstories in General (my Take)
Let's talk about backstory as a general concept. Backstory is the story BEFORE the story. I believe the backstory is completely theirs to decide -- as long as it fits into the world. If my world has no magic and they were raised by wizards, then I'll ask them to change some things. If their backstory includes a kingdom I don't have on my map, I'll add it. I may start the players far away from it, so I have to time to create a town that fits their backstory if there are some interesting things I'm not ready to deal with yet. I let a player write almost anything for a backstory to explain why they are what the character sheet says they are. They can invent people. The back stories should be written by the second session (or so) and shouldn't be amendable without talking to me. Because I will use the backstory if I get any.
I always say that it has to be from the character's perspective, and can't contain information the character wouldn't know. So, if they say in the backstory they give me "I was saved by a sea monster of some kind," I make clear to them that they have just given me the opportunity to pick the monster. If they say "I was saved by a leviathan," okay, they were saved by a leviathan, but I get to decide the WHY.
The only issue I've had is when a character the player wrote in their backstory shows up in the campaign and I have them betray the character -- which is fun, but the player can sometimes feel you've betrayed them. "Your governess who has always treated you kindly pops up from behind the obelisk, crossbow trained on you. They explain the reason they treated you kindly was because she's been biding their time..." etc.
Player 1 has a character who was saved from drowning by an unknown sea entity which left him with a cursed sword. Great! But now player 1 will occasionally try to send me ideas of what he thinks may have been the monster-- which is really not up to him.
Great. He THINKS it was X. Maybe he goes to look for an X, that it was a friend and may help him in the future. "The X snarls at him, laughing "I would never have done that. Apparently, you were mistaken about who or what saved you. But now tasty meat comes to me for a watery death, and my tasty snack."
If/when he objects, "You've made a lot of assumptions. Your backstory says 'sea monster', your character was sea tossed, you have no idea what it was that saved you."
Look at Critical Role, the way the Matt Mercer is using the same basic premise (bet it is where your player got it, and that might be influenced why the player is pushing in a particular direction) originally presented to him by Travis' backstory. When Travis said his patron was a sea monster with a "big yellow eye", Matt ran with it in ways Travis wasn't expecting.
Player 2 is playing a character that she played in a previous campaign, which I'm allowing because she was more or less expelled from the campaign by the DM refusing to do his job. It's fine, but her character has a developed parent figure that she's written out text RPs with (which I don't allow in my games, but others have.)
I don't get why this is a problem. Now the story has started, her mother is one of your NPCs. You can write her into or out of the story as much as you want.
Same Campaign Doesn't Mean Same Path
I'm running two groups with the same campaign, but both are at 7 sessions at the moment.
I hope you don't intend the two campaigns to run the same way. A printed campaign, for example, gives the players a world to play in. An overarching theme or goal, and people and places to interact with. It doesn't force the players through all the same hoops. It isn't a video game. Sure, the world is same, and players who have been through the same module might be able to share stories about how they handled the giant boss monster, but how they got there and how they handle it can be wildly different.
Player agency, player choices, player backstory might take two parties in the same world with the same goals down very different paths. When I'm writing a town, I write a lot of plot hooks for things the player MIGHT do. If they do, great, if they don't they are still there when they get back, only maybe the problem got bigger, or solved by someone else while the players were away. My favorite is when they party says "We'll come back and help the town later, we have this other thing we're doing." and when they return, the small problem has sprialled out of control and now a side quest that would have taken a few minutes of role play to solve is teetering on becoming a major story arch.