It would certainly be a problem if the next game refers to him as "Tommy" and the player doesn't know who that is. It's especially problematic if the player always names his characters "Arthur" and now there are multiple Arthurs in the world to tell apart. And it's especially, especially, problematic if the game universe includes stories in other media than Renpy so there's no good way to get the player's choices for the names of previous MCs.
Now, if the game is a one-off it's not as big a deal. Sure, the player could use a terrible name, like "Shithead" but that just hurts them, not everyone else.
That said, I've only seen games that let you name your MC, never the LI or LIs. It kind of begs the question, if it's more immersive for the player to use his own name, why isn't it even more immersive to have him going after his RL high school crush? And that begs the further question, why not let the player decide the circumstances of the LI and MCs relationship to further match his situation in high school? And that ultimately leads to, why isn't the player just writing his own story?
Tlaero
Right now, there's a rather interesting game suffering from that obsession with name change.
The game is
The List by
Uncle Loco . This game, as was explained by the Dev, is just the initial part of a multi-games story that "mostly" follow the same MC. So, it brings to the forefront the issue that you are talking about.
The MC we have right now, is going to go ahead and be the MC in further games, it comes to reason, that he should have the same name in the whole series. Since we can't foresee where the tech will evolve to in the future, if at the end of this rather large project, we will still be using Ren'Py or something else entirely, the most sensible thing would be to stick with the name the Author decided upon, so, no confusion arises.
But, alas, no. People want to name the MC after themselves. They continuously DEMAND in no uncertain terms, that they be given that right. No matter how many times you explain to them, that doing that is not ideal. They want it. They criticize the Author inability to allow them to change the MC's name. They asked a modder to create a mod to allow them that. It is so idiotic and so small minded that it boggles me that these persons are even able to breathe, much less read.
About the naming of LI's there are games that allow that. Then they try and go into the threads for those games and don't understand of whom others speak. Because ... nope, I got nothing...
I can understand a "customization" of an experience. Sure. When the game is pretty generic and does not rely too much on the characters, but mostly on what you do to them, OK, it can make sense to name your digital dolls. But, when the characters are an essential part of the plot, not as props, but as co-protagonists, I'm sorry... what are you trying to do? Re-write the story so it is about you? Is this what you mean by interactive?
Another concept that I rail against all the time is, Immersion.
This is a misnomer. People don't want Immersion, they want relatability. They don't want Reality, they want a representation of it, with which they can Relate. Experiences that evoke an echo in you, based on your past lived experiences. Because true Immersion, has was tested in several VR labs and abandoned, is absolutely unpleasant and no one could stand it.
What many people that claim Immersion want, is simply the feeling that "It could be Real". Even in scenarios where Fantasy and Science Fiction are the rule book.
I usually use the Default name, unless it is something idiotic like Chad or (very much) worse. Reading a story with me as the MC would actually bother me a LOT more.
Peace
