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I don't think Paizo ever intended on players being mounts. if you think about the various mounts Cavaliers and Paladins, and other archetypes can get- are mostly quadrupeds with basic intelligence- beasts of burden that aren't hindered by it's handler, when ridden into combat.
Ask yourself- If you're a vine, are you really "riding" an ally, or are you just wrapped around their waist or arm? More over, what's your dexterity/flexibility/cognisense while in your vine form?
As a vine, wrapped around an ally, i'd say you were more engaged in a "friendly/allowed" grapple- but even then, comes the hindrance of being grappled
Also, if you claim to "ride" an ally, i wouldn't be surprised if a GM starts making you do RIDE check when you try to do what you want to do.
As NorseWolf said, this is a possible cheesing situation: allowing a race with slower movement a "Free ride" on an ally to get full-attacks at no consequence of positioning.