People watch Too much television
Hey, wait a minute! I have a great idea for a new reality TV show, we'll call it "who wants to be a Chef?" See, we'll get this Chef--not just a chef-chef, but a Chef,Chef,Chef-you-know-a-real-Chef, one with full-sleeve tats and a knife made from a solid piece of unobtanium, forged by mystical dwarves on the very pinnacle of the Dutch Alps. Anyway, we get this Chef and give him/her a one year lease on a beat-up 50 seat diner and film the whole thing. If he opens on budget and can pay overhead within 3 months, we keep him, if not, we get another. If, after a year the Chef can pay himself a salary that is greater than minimum wage, we' ll sign him on for another 2 seasons.
But, being a reality show and all, we have to make it interesting, I mean, Chefs normally just cook and make a pile of money, right? Nothing interesting ever happens. But for every episode we'll throw a minimum of 3 curve balls, stuff like having the fridges break down when the health inspector comes, or "difficult" vegetarians who will only tell you what they can't have, not what they can. Oh! Oh!, wait! we can have cameos from like Jamie Oliver posing as a Groupon sales guy, convincing the Chef how selling your food at 75% below the menu price will somehow make you money, or Ramsey posing as a "professional blogger" conning the Chef to provide a couple grand worth of meals for a favourable blog. You know, stuff that never happens in real life.
Naw, the head office big shots will never go for it, it's not what the public thinks a Chef should be doing......
But in all seriousness, to answer the O.P.'s question, why do call themselves "Chef's"?
Same reason that bullies pick on little kids, same reason dogs lick their, ah, um...Private parts.
Because they can. No other reason
Look, if you want to call yourself a doctor, you better have a medical degree or a Doctorate's degree from some kind of a University.
Same goes for an engineer, a lawyer, accountant, or Architect.
Cops have to have a gun and a badge to call themselves a Cop.
Plumbers have to have a once-white van full of greasy tools and a dashboard full of invoices and Mc D's garbage or no one will take them seriously.
But you don't need
anything to call yourself a Chef. Nothing. No degree, no piece of paper, no badge, no gun, just a poofy white hat.
That's why....
Now, in continental Europe, a pro-chef doesn't call themselves a "Chef", they call themselves "Cooks". A "Chef" is what you call the Boss. My cousin in Switzerland owns a autobody shop, and his employees refer him as "der Chef", same goes for a niece who's a pharmacist, her staff call the her "die Chefin".
Only in N. America......