Interesting question. However, my answer is no. Vanity is not inherent to all competition. The key words here are "inherent" and "all competition." Therefore, one counter example would prevail via reductio ad absurdum. My chess counter example that comes to mind would be GM Arthur Bisguier. He was known for his friendly attitude when it came to chess. Some players criticized his "lack of drive" to win, but one thing for sure is that he was not a vain man when it came to competitive chess.
I think the more common observation (probably correct) is that a lot of players (maybe even most players) at the highest competitive levels of anything take on a more vain persona - but this isn't a requirement of the competitive nature: it is a side effect for many who take up the challenges of identity to labeling themselves by their trade.
Allow me to explain in another way. I've personally experienced high competitive levels for a lot of things. For example, I play golf competitively. It is one of the few things I began at a young age (there are home videos of me putting at age 2 or so and me playing in our backyard with plastic golf clubs around age 3). However, my father especially loved the game of golf as a game, but never wanted me to treat it as competition for fear (perhaps good reason) that I would lose the love for the game. My father learned golf as an adult, in a business setting of golfing after work. He was a solid golfer, but never great. He liked the game as a way to relax, but perhaps felt like he learned golf too late in life. He never forced me to play golf, but I think it was something I took up at a young age because I liked spending time with him and doing what he was doing.
Since I began golfing so young (and apparently gifted at it I've been told) I've worked my way through many competitive channels, but I've been careful to never lose that love for the game (and let vanity take hold). The result? I still love playing the game of golf! True that some events can become emotional or competitive, but in no way has vanity or competition dominated my golf life. Despite being at a high level, (in anything, but in my example it was golf) I can realize the ability of those around me. It is really competitive near the top! Of course, acknowledging someone elses' ability doesn't take away from your own: but I think it can sometimes keep one to retain humbleness.
Unlike some people though, I do not excel at one thing. Some people have mantras along the lines of "learn only one thing, but learn it well." I find this to be terrible advice if taken at face value! I believe that one should learn many things (perhaps as many as possible) to be well rounded and interesting, but that one should also take note of which things they love, or excel at; those are the things you should pursue and "learn well."
In addition to playing golf at a high competitive level, I've also excelled at billiards, art, psychology, magic, maybe chess and most recently I've been putting my love of psychology into Texas Hold'em Poker (and many other things I'm sure, if I think more on the subject). I've got a lot of things I work hard at - but I don't think vanity has encroached into any of these areas - even the most competitive ones.
I think that it is attitude oriented. If you can manage competition in a healthy manner, then vanity will not take over. Don't forget the classic Greek "Huburis." It is those that are higher that often fall farthest. It reminds me of something that sounds applicable here: "When pride comes, shame follows, but wisdom comes to those who are not proud." - Proverbs 11:2
In short, I don't think that vanity comes about in competition as an inherent property of the competition. I admit that many players do fall to this, but it can be prevented with the correct self-maintaining discipline. If you are personally concerned that your competitive nature will take away your love of chess, then that is important to note. However, the fact that you are considering this possibility now: tells me that you have the potential to improve your chess while maintaining humble. Typically prideful, arrogant players (in anything, not only chess) are the ones that won't even consider questions like you are now. If you think you are up for it, then I say: "enjoy you new chess journey and a game that most all of us here on chess.com love!"
Is there any competitively healthy way (i. e. any way which doesn't lead to a self-sabotage and/or auto-destruction performance-wise, if the results themselves should mean anything at all) to play chess without any vanity put into it? Sure, everyone's vain to some extent, but if one were to go about minimising it, how far could one go?
I'm asking this because I love chess, but recently I've become especially sensitive to my own vanity, so I wanna find a reasonable compromise between having the former and losing the latter as much as humanely possible. While every loss you experience can knock you down a peg so as to positively humiliate you, so can every victory exalt your ego to a height which, at the very least, isn't balanced and/or useful for a virtue-seeking human being.
I've already read about famous chess players and pro gamers saying stuff like "chess is vanity" and "not caring about the result leads to (competitive) self-destruction", so is vanity inherent to all competition, or not? Is caring about personal skill improvement in a sports game like chess inextricably tied to the desire to prove personal superiority by "crushing another person's ego", in Fischer's words? Is that always the main reason why we play chess, be it the only one or not? Or is there a way to remain saintly-minded while becoming better at chess over time, however slowly?
I would like to think that competitive improvement doesn't necessarily become ego food, at least not when experienced moderately enough... but I can't say. What are your experiences and opinions on this matter?