She originally showed up in an early season 1 episode and was considered a villain at this time. She made a return in season 3 in which she takes on a far more villainous role, until the Alicorn Amulet is taken away and she makes peace with Twilight. From season 6 onward, she has some more regular appearances including interacting a lot with Starlight Glimmer. Ships of them were born out of this. I don’t recall how much of a role she played in Equestria Girls, but she did make appearances there. Given magic is much less accessible in that setting, being a stage magician would stand out a lot more.

So, here’s the thing. Trixie was popular long before season 6. Hell, she was probably popular before season 3. That’ll mean she was in just one episode, not counting any background appearances. Equestria Daily really seemed to be into Trixie, which is mainly what I’m basing this thread off. There was even a lot of Twilight and Trixie shipping including two different webcomics on both characters being genies. So oddly specific.

Personality wise, she’s a stage magician in a setting where actual magic is normal. Her magic is also real, it’s just geared toward performances and making quick getaways. She also speaks in the third person a lot. In fact, in most fan fics I’ve read where Trixie is in it, there’ll be something that happens that gets her to refer to herself as I or me instead of her name. I wonder why this happens so much.

What makes Trixie so popular? Her personality? Her looks? I’m not trying to downplay any of it, I just don’t know. And one more thing. Was her full name, Trixie Lulamoon, ever spoken in an episode or was that displayed on merchandise only?

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My theory is there's a lot of narcissists in the community and they see her as a pony they can relate to.

You know the kind. "I have to see myself being represented by the characters in the production I'm watching or I can't possibly enjoy it!"

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What makes Trixie so popular? Her personality? Her looks?

I say those factors are the reason the character became popular. Even I ended up liking Trixie myself, mostly her personality and her desire to impress the ponies despite not performing exuberant magic spells.

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Trixie was never a villain, she was an antagonist. It made her redemption arc and eventual hero turn a little less jarring.

There has been a few times where she's referred to herself in the first person in the show. In Forgotten Friendship, she lends herself to a comedic scene when she executes the magician's exit. The scene is even sillier when you realize that the room where they're doing yearbooks locks from the inside.

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Because she’s blue!

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One of the reasons is that a number of fans didn't like how she was treated in "Boast Busters" and felt she was treated unfairly or even wasn't in the wrong at all. Few things help boost s character's popularity than them getting shafted somehow. But another reason is her boastful personality, which can potentially make her a fun character to watch even in season 1, let alone season 6 and onward where she had mellowed down a bit. Although her boastfulness is sorta a double-edged sword, since it repels some fans from her, as well as making her more charismatic to other fans.

And yeah, her looks probably didn't hurt either.

On a side note, season 6 Trixie and beyond doesn't refer to herself in the third person nearly as much she does in a lot of fanfics during or past season 6, including many of my stories with her. In canon, season 6 and beyond Trixie refers to herself in first person most of the time and usually only situationally talks in third person, and this generally carries over to her human counterpart too. For example, these are scenes from "No Second Prances".

Starlight: You know each other?

Twilight: You could say that.

Trixie: We've had our differences. What matters is Twilight gave me a second chance, and I appreciate it.

Twilight: So, um, what brings you to Ponyville?

Trixie: The Grrrreat and Powerful Trrrrixie has come to perform a new stage show of grand illusion! I am calling it "The Humble and Penitent Trixie's Equestrian Apology Tour"!

Starlight: That's kind of a mouthful.

Trixie: It's a working title.

Starlight Glimmer: What is it?

Trixie: I heard what Twilight said about me, and she's right. I wasn't very nice. So I'd understand if you didn't want to be friends.

Starlight: Are you kidding? You're the first pony I've met who has any idea how I feel!

Trixie: Can you keep a secret?

Starlight: What are friends for?

Trixie: The things I've done? I did them because I was jealous of Twilight. She's just the best at everything, and I wanted to beat her at something!

Starlight: Your secret's safe with me.

Trixie: Thanks. Want to help me unpack my wagon?

If that last scene was a fanfic, depending on who was the writer, it probably would have looked something like this:

Starlight Glimmer: What is it?

Trixie: Trixie heard what Twilight said about Trixie/her, and she's right. Trixie wasn't very nice. So she'd understand if you didn't want to be friends.

Starlight: Are you kidding? You're the first pony I've met who has any idea how I feel!

Trixie: Can you keep a secret?

Starlight: What are friends for?

Trixie: The things Trixie has done? She did them because she was jealous of Twilight. She's just the best at everything, and Trixie/she wanted to beat her at something!

Starlight: Your secret's safe with me.

Trixie: Thanks. Want to help Trixie unpack her wagon?

Even in "Magic Duel", Trixie spoke in first person at a surprisingly high rate, but she also spoke in third person a lot too.

Trixie: You two! Hurry up with my throne. And you, how long do I have to wait for my applesauce facial?

Applejack: Forget it, Trixie! I ain't doing nothin' 'til you let Twilight come home! [laughs uncontrollably] Okay, I'll do it! Just make it stop! Hoo!

Trixie: I thought I told you to dance!

Trixie: This better not be another false alarm, or the Great and Powerful Trixie will– [gasps] You! What's the matter, Twilight Sparkle? Not enjoying your exile?

Twilight: I know about the Alicorn Amulet. I know you cheated.

Trixie: Cheated? Moi?

Twilight: Yeah. And I thought you might wanna see what a real magical amulet looks like. Zecora gave it to me. [muffled] It's from beyond the Everfree Forest, and it's way more powerful than your measly little Alicorn Amulet!

Trixie: Ha! Nothing's more powerful than the Alicorn Amulet! And nopony's more powerful than the Great and Powerful Trixie!

Twilight: [muffled] Care to put your amulet where your mouth is? How about another duel?

Trixie: Why should I? I already beat you.

"Boast Busters" seems to be when she spoke in the third person the most and is closer to how fanfics often handle it, but even then she used first person a few times.

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I wouldn't rule that out for at least a few people.

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It's easy to get the two mixed up. At least in season 1, I couldn't tell you exactly why she gets considered a villain. Her appearance in season 3 is much clearer, but that mostly only seems to be tied to the Alicorn Amulet. Once that's out of the picture, she asks Twilight for forgiveness and I don't think she ever took on another antagonist role from then on at least in FiM.

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So I see this was canon for quite a while which might be what inspired fan fics to do the same thing.

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Well to be honest, I never seen her as a villain. I mean I know that some performers tend to get heckled by arrogant audience members and they tend to embarrass them for mocking them when they’re trying to entertain others and gain some payment for their performance. I mean I don’t understand why the entire town blame Trixie for the Ursa Minor when she didn’t actually brought it in town. Also for what their actions against Trixie, how come other performers like her didn’t show some anger towards Twilight and her friends?

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how come other performers like her didn’t show some anger towards Twilight and her friends?

Ponyville might not have had other performers that acted like Trixie did during their performances and thus had little reason to empathize with Trixie that much; they could have even thought she was a jerk too. It's also possible pony performers don't think the same way as real performers do, so human logic might not apply to pony performers.

And as for performers outside of Ponyville (assuming we can apply human logic to them so they would be offended or angry by Trixie's treatment) then the reason they didn't is likely because other performers either never heard about it, didn't get the full story, heard a skewed version of it (happens a lot in real life), or just never met any of the mane 6 in person so they couldn't show anger at them even if they mad on Trixie's behalf. It's not like many (if any) real-life performers, given they could even verify what they heard was true and not gossip or rumors, would travel halfway around the country just to tell off a few jerks that heckled a follow performer that they had never even met, let alone form a strong enough bond with said performer to go this far. Equestria doesn't have the internet, so Equestrians can't just jump on their version of Twitter or whatever and rage there over something they heard like people can do in real life.

EDIT: But I wouldn't be surprised if a few early fanfics addressed this, but they fell out of favor years ago.

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She did get a minor reprisal as an antagonist for one episode when she and Starlight bonded over their mild animosity towards Twilight.