You can tell a lot about a person from the compliments they give. Karl Lagerfeld, king of fashion, likes to keep it niche. During our interview in a central London hotel room a couple of his actual friends pop in amid the assistants and PRs. One is Lady Amanda Harlech, his long-term muse and collaborator, whom Lagerfeld introduces rather gallantly as "an English inspiration". The other is a fashion editor whose hands Lagerfeld guides in front of my face as he barks with gusto: "Look, she has the hands like the 17th-century painting by Van Dyck, no?" Like most of the designer's pronouncements, he doesn't really require an answer, but I'm forced to agree – her hands do have something of the Henrietta Marias about them. It's an odd compliment that manages all at once to be grandiose, cultured, smart, silly and entirely on the money. Much like the designer.

Lagerfeld's first eponymous London store.

Lagerfeld's appearance in the flesh lives up to expectations. The collar is high, the hair is powdered, the lips are huge, the glasses dark. His conversation has the rhythm of an electric drill bit that has unexpectedly hit a section of concrete. He spatters out soundbites – tat tat tat – in a heavy European accent. When he's finished with a topic, he claps his leather fingerless gloves together with a hollow thud. As if to say, done, move on. Like his appearance, his conversation style is designed to protect his privacy. He throws out insults ("to be sloppy after 25 is a problem. You have to be really fresh to wear stuff that doesn't look fresh"), opinion (on tattoos, "it's like sleeping in a printed T-shirt, you will never get clean again") and faux-fashion-baddie stuff ("tell me, why do former young designers who are now in their middle 40s have to redo the 1960s and 70s. Why they cannot invent fashion for today?") Cara Delevingne, one of Karl's favourites, also gets a 17th-century compliment, for her unconventional browline: "She is not a standout beauty. Was it Bacon who said, 'There is not beauty without some strangeness in the proportions'?" Meanwhile, the cult of the selfie gets short shrift: "They are this horrible thing where you are distorted. The chin is too big, the head is too small. No, this is electronic masturbation."

Cara Delevingne walks the runway at Lagerfeld's supermarket Chanel show at Paris fashion week. Photograph: Dominique Charriau/WireImage

What the Lagerfeld soundbite machine amounts to is an effort to get people to focus on the tantalising titbits and ignore the real man. But it is all done with humour and a surprising amount of warmth – his lips constantly battling not to turn up too much at the sides. In the flesh, his conversation is not mean or diva-ish. It's meant as entertainment – an act of self-parody on an international scale.

At 75 or 78 (depending on who you believe), Lagerfeld is widely viewed as a genius of modern fashion and right now he is at the very top of his game. He's late for the interview, naturally. Two hours late – his jet is still circling above a private airfield outside London as we're due to start. He's in a rush, too, although his politeness doesn't give that away. He has popped over from Paris to host a dinner to celebrate the launch of his first eponymous London store and, despite the two Chanel-branded suit bags he has brought with him, he's not staying the night. He's got to get back to Paris before the design studios close for two weeks, then sign a contract for a hotel he's doing in Taiwan. But he's charming and courteous and says of his infamous workload (which stands at 17 collections a year and includes designing for Chanel, Fendi and his own line): "If you like what you do, you don't count."

Lagerfeld with his cat Choupette, the ultimate social-media marketing tool.

Lagerfeld has built the business model for his own brand empire by embracing self-parody. In his new shop, everything is designed around Karl or to make the wearer look like Karl. It's intentional, of course. "Why not? It's fun, no?" he says. There are removable Karl collars on sale for a little over £100, sunglasses that somehow make the wearer look like Karl (I can personally vouch for this) and tokidoki dolls complete with his cat Choupette. The cat (which is quite James Bond baddie to complement the designer's mock evil-genius vibe) is so perfect as a social-media marketing tool that I can't quite believe that it exists in feline form. But Lagerfeld claims he gets picture updates of Choupette's activities from the cat's maid every hour. As proof, he shows me the most recent one of her resting on a pile of books at his home. "Everything is controlled by her. She wakes me up at 7am because she wants me to bring her fresh croquettes, she won't touch food from the night before, she gets offers for food commercials but it is out of the question. She is a kept woman." He turns the smile down and swallows a laugh.

Lagerfeld's retro-styled trainers, more affordable than their Parisian cousins.

Right now, Lagerfeld is into trainers. It was the shock styling tactic he used at the most recent Chanel couture show, putting tweed trainers with £40,000 outfits. At his ready-to-wear show for Chanel earlier this month he created a parody of a supermarket, with models including Cara Delevingne and Stella Tennant in holographic trainers, emptying the shelves of Chanel-branded cognac and feather dusters. In the new Lagerfeld store, a whole wall is dedicated to Technicolor retro-styled trainers with a K initial on the side that resemble the New Balance pair that have been breeding on streetstyle blogs over the past 18 months. Crucially, the Karl trainers, like much in his shop, are more affordable than their Parisian cousins. Lagerfeld is clear about the business logic of pricing his eponymous brand below the two other lines he designs for. "You cannot dance at three parties on the same evening. I always said to my investors to put it on another level and make it affordable. I think that's a modern attitude."

The trainer fascination is typical Lagerfeld. He seizes the zeitgeist with humour, throws in a historical reference and makes the look his own." Everyone is wearing them," he says. "I thought it was an elegant idea to put them with dressy things. If you look at fashion from the 1840s women wore flat shoes. Stella Tennant, in real life, I've hardly ever seen her in stilettos." By putting his high-end clothes with trainers and making his models do normal things such as supermarket shopping, Lagerfeld shone the spotlight on the way we admire stylish women today – more likely three metres away than on a catwalk.

Mini-Karls: Lagerfeld tokidoki dolls, complete with cat.

He also inadvertently turned fashion editors into shoplifters. As the show finished there was a rush for Chanel-branded supermarket "souvenirs". Did he realise that he would create an haute scrum? He raises his eyebrows high. "I saw it on film later and the bodyguards were removing things from people's bags. I didn't expect that of people who normally come to my shows." The consumable stock, he says, was intended to go to homeless shelters, the empty packaging was destined for the Chanel window displays, not editors' handbags.

Despite such hunger for Lagerfeld and his relentless imagination, the man himself is never satisfied. "I'm never happy and pleased with what I'm doing," he says, which is good news indeed for fans of his grand artistic gestures. He already knows how he will follow up the supermarket he says, although he won't be drawn into discussion other than to say that couture will be "chic" and the next ready-to-wear show will be even "wilder". "I always think I'm lazy, maybe I could do better, I could make more effort and I always have the feeling that there is a glass wall that I cannot get through. But maybe when I get through, then it's over."