I'm going with what Wikipedia describes as Mary Sue. If you don't like it, tough shit. Post all the YouTube videos you want.
Wikipedia Link
The term "Mary Sue" comes from the name of a character created by Paula Smith in 1973 for her parody story "A Trekkie's Tale":15 published in her fanzine Menagerie #2. The story starred Lieutenant Mary Sue ("the youngest Lieutenant in the fleet — only fifteen and a half years old"), and satirized unrealistic characters in Star Trek fan fiction. Such characters were generally female adolescents who had romantic liaisons with established canonical adult characters, or in some cases were the younger relatives or protégées of those characters.
She's established as the foster sister of Spock. There's nothing inherently wrong with that but for the sake of argument, I'll give you that one.
By 1976 Menagerie's editors stated that they disliked such characters, saying:
Mary Sue stories—the adventures of the youngest and smartest ever person to graduate from the academy and ever get a commission at such a tender age. Usually characterized by unprecedented skill in everything from art to zoology, including karate and arm-wrestling. This character can also be found burrowing her way into the good graces/heart/mind of one of the Big Three (Kirk, Spock, and McCoy), if not all three at once. She saves the day by her wit and ability, and, if we are lucky, has the good grace to die at the end, being grieved by the entire ship.
Burnham didn't join the Academy at a young age, and seems to be a normal age as far as Star Trek characters go for where she is.
"Mary Sue" today has changed from its original meaning and now carries a generalized, although not universal, connotation of wish-fulfillment and is commonly associated with self-insertion. True self-insertion is a literal and generally undisguised representation of the author; most characters described as "Mary Sues" are not, though they are often called "proxies" for the author. The negative connotation comes from this "wish-fulfillment" implication: the "Mary Sue" is judged as a poorly developed character, too perfect and lacking in realism to be interesting.
Unless you believe Michael Burnham is a representation of Bryan Fuller, she's not a self-insertion.
She has character development. We know her personality. We've seen her relationships. We've seen her warm up to characters, particularly Tilly who she was standoff-ish with at first and Saru who she used to always butt heads with.
She's also had to come to terms with the fact that Sarek is not as perfect as she thought and has to deal the realization she she spent years thinking she wasn't good enough for the Vulcan Science Academy when in fact Sarek just lied to her. She was very Vulcan at the beginning of the series, spent time in prison which hardened her, then was on Discovery when she began to decompress from what happened at the Battle of the Binary Stars and from the six months after, then when she found out Sarek wasn't upfront with her, she started to feel okay to grow more distant from being Vulcan.
She's on a discovery of what it means to be herself and not a Human Among Vulcans or Career Starfleet Person, since she didn't believe she'd have career beyond however long Lorca needed her. She's been trying to discover being just Michael Burnham. And she's getting used to Discovery and its crew which is different from the Shenzou and its crew.
On top of this, we're only 13 episodes in. For 13 episodes, there's character development. Whether or not it's character development
you like is a different story.
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So, other than being the foster-sister of Spock, Michael Burnham fits no definition of being Mary Sue. She's just a character that some of you don't like.
On to the TV Tropes Page, which I'll compare against the Wikipedia Entry on "Mary Sue".