Thanks for sharing! Makes me feel a bit better about the situation 
As for cousins choice of names, it does sound a little strange, but there is a reasoning in the story for that, which I can't say cause it might spoil it for some people
I totally agree with you. Problem with me is that whenever someone tries to tell me they don't like something I do, it gets stuck in the back of my head and annoys me for days , I know that I shouldn't be doing that.
But I'm glad that it doesn't bother others
I think your classmate needs to recheck the definition of a Mary Sue first.
A Mary Sue is a type of fictional character, usually a young woman, who is portrayed as unrealistically free of weaknesses. Originating in fan fiction, a Mary Sue is often an author's idealized self-insertion.
To me, it sounds like they are jealous 
I don't mind self-inserts, they are bound to happen, but I think we all, or at least the majority of us, dislike Mary Sue characters.
Also for the naming thing, where I am from, it's actually common for cousins to have the same name if the families have a lot of kids and they maintain close relationships. At large family gatherings I attend, I am sure there are at least 4 Christophers on the dinner table. It was my grandfather's name. I see it less and less with the children my cousins are having (as in the next generation) but the generation before us did it often from my personal experience 
Well, you got the jealous part right
This classmate was the type of person who would do anything to make themselves look better than everyone else. They are an art student, and they didn't like people who took their limelight ( especially other art students such as myself.)
As for the topic of names, I actually did the close relationship idea with Natasha and Tasha. They are extremely close, have each others back and do stuff together etc. They are almost like sisters.
I wouldn't do it, to be honest. Even if you don't mean anything by it, you might give off the impression that you do, so I'd rather just avoid it altogether, save myself the trouble of explaining.
I do have a few characters named after me. Though, I don't use the direct spelling.
In my eyes, all of my characters have a piece of me in some way. And if they don't, the events they go through and their stories' themes are somehow related to my beliefs and developments.
So a character sharing my name isn't really much to come off egotistical. I've already put myself out there on main before doing so.
I have seen it done before but sometimes it can be a bit odd.
There was a comic where the author and MC were named Frank. The character was a giant loser and he had a crush on a sexy woman. Stuff like that sounds like possible self interest to me. It can also be distracting and makes me question a author if the character does something questionable.
But then you have Star Wars created by Lucas with a MC called Luke. The two are very different but it is sort of funny how similar the names are.
For Adventure Time, Finn’s original name in the pilot was Penn, named after the creator Penn Ward. He had concerns how the character could be used down the line and decided it might be best to rename him.
I mean do you, if you want to name a character after yourself then do it.
Honestly the thought of naming a character after myself has never crossed my mind but I personally would never do it thought. Just because I want to draw a clear line between my work and myself, but that's just me
I don't really reveal my real name online publicly so if I named a character after myself, (almost) no one online would know. 
Still I'm not a fan of my real name so I probably would never want to use it for a character, lol.
I don't think there's anything wrong with giving your character your name, though! It's not even like it's a self-insert in this case, and even in cases where it is a self-insert, there's a place for those in fiction too, and no one should be shamed for writing things that some people might deem "cringe".
When you said a name similar to her cousins, I immediately thought of Tanya!
But Natasha is a pretty common name in some parts of the world, so if it works, then I don't think you need to change it.
Personally, I would never name a prominent character my name, mostly because I'm just lukewarm on my name. It's not a very common name too so it raises more of an eyebrow. I might use parts of it, like "Jo" or "Anne", but having it exact would just invite opinions from other people that I'd rather avoid entirely.
I don't see anything wrong with it, in fact quite a few popular comic creators insert themselves in their own stories, some names include Adamtots and Litterbox Comics. Adamtots puts himself in his own comics from time to time and Litterbox Comics is literally based on the creators life as a mom, but her and her family are these fun cat characters. And if I remember correctly, even Stan Lee occasionally inserted himself in some of his works (feel free to correct me if I'm wrong).
Heck even I sorta did that for my comics since a lot of these creators inspired me I even based a few of my characters on people around me, as well as having a sorta persona of myself in the comic. so to me it just sounds kinda like they're just a little jealous, don't let it get to you, just keep working on your comic and ignore them. 
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Yeah, I see what you mean. When it came to naming my character, I didn't give her the version of my name cause I knew it would raise a few eyebrows. Since Natasza isn't as common as Natasha. And the mispronouncing of the name would also be a huge problem. Trust me, I know
I hope I helped hehehehe
Just know its always gonna be a bit cringe when reading back your own work to yourself or someone else.
Nah you're fine haha. I went a step further and used self & friend OCs that I made back in highschool of me and a few of my buddies as the main characters in my first one-shot 
Decided I just wanted to make a comic right away and didn't want to spend a lot of time developing new characters so I just updated the designs and tossed 'em in.
That said, I wrote them as their own characters for the story, only very loosely based upon personality traits and such of their inspirations, and although I have my name on the cover which is clearly the same as the MC, no one ever commented or asked about it... except the friends in questions who were the other characters, who all thought it was awesome and thoroughly enjoyed having their personas in a comic xD
I did decide to stop the comic after the initially planned arc, though, both because I felt limited in how I could portray the friend OCs in some ways, as not to show them in a negative light or something, and I was also feeling a little self conscious afterwards about having a self-insert MC as well even though I felt comfortable doing anything the character arc or plot required with that character even if it wasn't flattering, because idgaf 
Naah~ My MC is Seth (like myself) in addition, their game ID is Sethstiel hahhha 
Although they're nothing like me ^^
I see it as it's just a name. I like my name, I think it sounds cool so why not naming them that? 
I really don't see why it would be a problem. I only don't do it with the main characters that because I don't feel like, but I always keep a background character similar to me and bearing my name in mystories.
Also one of the most famous characters in fiction share a name with his creator. LUKE (Skywalker) is just a variant spelling of (George) LUCAS