What is a Mary Sue?

An unrealistically perfect character with no visible flaws?

A character who receives undeserved adoration to the point that the other characters become mere extensions of them?

Some combination of the above?

Does Mana Aida of Doki Doki Precure fit these qualifications?

When the show first aired, the fandom was quick to label Mana a Mary Sue simply because of her extreme desire to help people around her with little provocation. 

At this point I can kind of understand since she doesn't seem to have much of a personality other than a desire to help others, but it was still no reason to call her a Mary Sue, especially since early on (like episode 5) she showed that she has some genuine flaws, like being over enthusiastic to the point of not considering other people's feelings.

Then Regina walked into Mana's life and everything just got weird. When Regina shows up one day declaring that she wants to be friends with Mana (out of sheer desire for amusement mind you, not because she actually cares about Mana), the latter decides she wants to be Regina's friend too.

Naturally her friends are quite aggressively against it, and they should be. This girl is a complete amoral sociopath who would destroy entire buildings and kill people just to get her own selfish, bratty way. She's also the daughter of the person who ruined Makoto's world and doesn't feel an ounce of remorse about it.

Not only that but Regina constantly mistreats and talks down to Mana's friends, but does Mana care? Nope. It's not until Regina tries to outright KILL Mana's friends that the latter steps in and says something, and what does she say. Something along the lines of "I still want to be friends with you, but you need to understand true friendship."

And why does Mana want to be friends with Regina? Basically all we get is "Because my instincts are never wrong." And not because Regina has shown any signs of wanting to be a good person or being lonely and misunderstood.

In general it seems that Mana brings out the complete stupid in Regina. When Regina gets brainwashed and forcibly pulled back to the dark side, Mana seems completely oblivious of it, even willing to rush over to her side and chat with her (something Regina actually uses to her advantage once)

It takes a while, but somehow Mana manages to get all of her friends to one-by-one express feelings of friendship toward Regina through things such as song and flowers. In fact we're never given the impression that they value Regina as a friend. It's more like they're just doing it because that's what Mana wants.

Doki Doki Precure seems to ooze yuri undertones, but the one thing people will notice about it is that the vast majority of it is directed at Mana, leading to the fandom joking that she has her own harem. This harem collecting extends as far as Miyuki in New Stage 2, and no doubt will happen with Megumi in New Stage 3. 

Time and again the show has proven that Mana is the center of the show's universe, and it's been confirmed by the writers that they have the most attachment to her. 

She is ALWAYS the one who uses the finishing attack against the monster of the week. The other Cures will often give their power to her in a critical situation, instead of uniting their powers together. 

Heck, Mana is the Cure I've ever seen to take on an actual monster of the week by herself, while de-powered, using only her voice an an umbrella. And this monster was creaming all three of her transformed friends.

And don't even get me started on the final battle. Mana gets her heart forcibly tainted by selfish energy and it bursts out, just like any other monster of the week victim. And then... it just changes back. It just... un-taints and goes back inside her. No explanation of how this is possible.

It would be understandable if this were someone who is shown to be absolutely purely selfless, but Mana is not such a case. Maybe it's just the way she words things, but she seems to do things to help others because of her own personal gratification. 

"It makes me happy when you sing." 

"Helping others makes me feel so warm and happy."

It puts her whole thing with Regina into perspective, like she wants to reform Regina for herself, so she can feel good about changing someone's life, not because she sees any actual good in Regina.

And then of course there's the absolute final battle of the anime, where the other Cures once again send their power to Mana, and even Regina joins in, all so that Mana and Mana alone can get the final power-up and fight the final battle herself.

This kind of thing seems like it belongs in a shounen fighting anime rather than a Magical Girl warrior team story. And it makes sense because the director of this series worked on One Piece.

Does all of this make Mana a Mary Sue?

short answer: yes. 

long answer: when i think about Mary sues, I usually think of bad fan-characters (male and female). 
you know the ones.

where they come in out of nowhere, are basically dream-self-insert characters of their creators, are instantly the highlight of all the characters worlds with little to no time or effort, and have all the super cool powers and privileges. 

I think the best example of this I've ever seen in something canon is Noike from Tenchi Muyo (any tenchi fans in here?). Its a harem show and she's introduced in the third series, comes the heck out of nowhere and announces shes Tenchi's legal fiance, is adopted royalty, has the biggest boobs (and in harem anime this is important), is a high ranking galaxy police officer, can cook really well, and becomes the sort of mother figure for the rest of the girls who are spending the entire 3rd season in the sidelines. 

the only thing that holds Mana back from being a "canon-sue" is that she was already the shows main character from the get go, so of course the show revolves around her actions. But it comes off as kindof obnoxious because Mana doesn't really learn or change. She's the exact same person at the start and end of the show. 
I think someone had the goal of making her the ultimate role model and it just went awry...

I admit to being one of the people even early on who called out her over the top perfection (she's so awesome at being student president, the teacher can take a nap on the bus during a field trip!) but atleast she was interesting to watch cuz we spent a little more time in her head. 

After awhile I just kinda felt indifferent to her because the show just straight up told me "mana's not perfect all the time" and didn't give me any evidence to back it up so it was just annoying.

You can have really interesting characters that are well-meaning, cheerful, justice seekers (see Utena Tenjou or Honey Kisaragi) but there's no depth to Mana's actions at all that its just boring to watch. 

and that viewer boredom then becomes character hate. and that's how Mana ends up as a "Mary Sue"

I liked Mana at first and I will admit I didn't get why people were calling her a mary sue. Now I see why. She really doesn't change liked Link said. They did write that she didn't understand people's feelings, but sadly they went no where with that. If they kept up with that flaw then maybe Mana would have been a fleshed out character. Yes it's good to help people, but you need to consider if they want your help or not ( That's a good lesson for kids). It also would have help if she had true reason for wanting to help people. Most selfless people do it to make people feel good and make the world a better place. I did not get that vibe from Mana.

As for Regina I didn't get why Mana wanted to be friends with this girl. She nearly killed her friends and her father destroyed one of her friend's home. And she was clueless about why Mako was upset? 

I kind of wish Mana was out of picture. I liked the three girls. I have no opinion on Ace since I stopped watching after her appearance.

No.

Doki Movie spoilers below add more fuel to the fire that one could claim Mana is a Mary Sue.

Spoiler:

Doki would have been so much better if the other characters were allowed to flourish on their own without orbiting around Mana.

Gosh! I never saw the movie. But yeah that makes Mana more sueish.