It is totally legitimate for a GM to accept/reject characters/rules/etc in his game.
Some players might argue/question the decision but in the end, this should not cause offense unless you are very clumsy in how you announce and/or explain it.
Personally, I recommend that you discuss why you do not want to allow it and your players should respect that. As the GM you have the final say in everything but, in my experience, a GM that simply puts his foot down without any explanation has a good chance of simply antagonizing his player(s) who might leave to find another table.
If your player(s) argue it might make your job more difficult, but you should arm yourself with patience, at least until the point where your player ends up being stubborn and unable to compromise.
Since your goal is to make sure you don't antagonize your player(s), I feel like you will have to explain the how and why you do not want to accept that character in particular.
It is difficult to give you examples/arguments to answer this question, since you do not expose any of the reasons why you do not want to accept this character in your game.
Is it because of the style of the game, you feel like one of those characters do not fit in a Gothic Horror kind of game ?
Is it because it is not appropriate, being a homebrew class that you do not like ?
It is because the theme of the character does not fit in your game ? As an example of this, I once wanted to play a cleric with a GM I had never played with, not knowing until session 0 that his homebrew world did not have any sort of divine magic in it, the Gods having left this world a long time ago ... the GM and I had to have that discussion where he explained this to me, where I tried to have see if he would allow a 'different kind' of divine spellcaster that would get his spells from a different source in this homebrew world. In the end I was not allowed to do it, but since the explanations given made sense about the kind of game the GM wanted to run, I had no issues with making another character more fitting for the game. Even more than that, my character was later allowed to multiclass up to level 3 in a divine spellcasting class because of what he did during the game ... since it ended up being revealed that the background story of the campaign was exactly about the Gods trying to make a comeback to this world, so it turns out I just could not start as a divine spellcaster (and, in the end, I did not multiclass into cleric since what the Gods did to return to this world kinda 'turned me off' both OOC and IC, lol).
The question has been edited to specify that magic will be rare and violence will be commonplace, while the player wants to play a pacifist illusionist.
Now I see that you are in an extreme situation. Not only the concept behind the character does not fit at first glance but also the class chosen is one you wanted to restrain somewhat in this setting.
I don't know if you were ok with a wizard in your group or if the game only allows for wizard NPCs and I'm really on the fence with my answer on this one. I'd personally be willing to go both ways, both as a GM and a player.
There's no good answer and in my opinion it will come down to a compromise.
On your part as a GM are you willing to still allow a character that could be 'contrary' to your setting ? Either allowing a pacifist in a violent world or a wizard in a world where magic is stygmatized both can be very fun and rewarding if done correctly and that depends both on the player, GM and the rest of the group. Or maybe you have something else in mind and would rather not bother dealing with that ?
On the player's part once he realizes this about your setting, does he still want to play that character concept ? Is he comfortable with the consequences of using magic publicly in this world ? Is he comfortable seeing his pacifist character getting abused by violent thugs ? Or would he rather play a simpler character ?
Check if compromise is possible for either of you two, then once you have a better idea of each other's point of view, the matter might be settled or you might need to talk about it with your group ... for example, if you decide to allow such a disruptive character, you might want to discuss this with the rest of the group since they will also have to deal with the impacts of this character !
Good luck with your group !