this is the weirdest comparsion ever no offense. So here is a complete list of AP assasins who youw ill fluently see mid lane. If anything katarina is the most unique of the ap assasins.
Katarina is a champion who quite literally has zero utility from passive-R, no utility, no utility at all unless you put a 
on her and in return, she sends out insane amounts of aoe damage. When played to perfection she is possibly the most mobile out of all the assassins, and posibly deals out the most hurt. Core weakneses being, very easy to cc, and very trash lane phase. She's a very good champion for diamonds and challengers to smurf on, for if they know how to play her at least on a diamond 2 level, they can hard core pub stomp games and end them fast. If I had to compare Kata to anyone it woulden't even be the AP champions but instead 
or 
simply because they also are pure damage/mobiltiy and no cc with the exclusion if their very soft slows.
So to answer your real question though? What is ahri? Ahri is still an ap assassin. Frankly it triggers me so hard when people call her a mage, for even Ezreal one of her designers calls her an assassin. An assassin has mobility a mage does not, that's where one of the clear differences is. But before I get to off topic. Ahri is indeed an assassin. I will adress each point you made one at a time.
1-"she can't burst anyone unless shes full build"
False: an ahri if she has all her abilities up and is keeping pace with her teamates and the enemy can easily ult ontop of the enemy adc or mage player and 100 to zero them, just unlike other champions she quite literally is not allowed to miss a single ability, to miss even 1 ability can result in the target leaving. Landing charms is critical and a must. So I agree she does lack damage, but not to the levels you say.
2-"I often hear ahri is mobile"
Reply: She is indeed mobile, with her Q speed up and her R on a somewhat fair, normal level cooldown, she can be quite the tricky fox, while I do agree with you, many people over hype her mobility, for champions like kata or Fizz greatly out class her in the mobile department, people on the other hand at the same time underestimate it, most my ahri kills are from people not taking my ults into consideration. Her mobility is what i like to call "it's decent" range, nothing over the top, nothing under performing. If anything the people who say she's "to mobile" are junglers who can't figure out how to gank her properly.
3-How often do you see a ahri solo carry a game?
Reply: How often do you see any champions solo carry games? Riot has unfortuantely in my opinion decided to over nerf every mid laner in the game, not just assassins but mages to. They were tired of mid laners being the stars in both lcs and solo q and they spanked us for it. They were tired of the "mid lane is the most important position" belief and did massive nerfs upon mid laners across the roster, assasins, mages, mid lane adcs, all nerfd. Honestly if a mid laner solo carries a game at this point, I don't care what champion he was on, it was because he was good and deserved to win.
4-"kata better at team fighting"
MEGA FALSE: Kata has no cc, sucks at killing tanks, is horrible at sieging, she's good at one thing and one thing only, punishing people out of position. In high elo she is so fucking hard to pull off, for any support or tank player at that elo knows "if i cc katarina, she dies". Ahri shines because she is so great at the things katarina sucks at. Sieging she can clear minion waves and harrass with Q, she can zone people away from turret with her charm, in team fights if shes smart, she can find the adc or major threat and stun them with her charm, so her team does not need to worry about them. People pick ahri for she blends so well in every comp unlike kata who says "pick a comp around me or else".
Q: what is ahri?
An ap assassins who excells at sieging towers with the functionalities of her Charm, someone with a great lane phase and powerful roams, but should not be expected to hard carry into the late game.
