There's an obvious freshness to Grayson Perry's evolving look, which is getting more sophisticated as time goes by. The chic navy outfit complete with showstopping hat he wore to receive his CBE from Prince Charles was deemed to be 'entirely appropriate' by Palace officials, while today at his pristine studio in north London he's in his trademark blonde bob with a short cerise dress (short enough to reveal matching cerise knickers) and clumpy platforms.

And there's nothing but freshness when it comes to his crowd-pleasing, debate-provoking exhibitions, like his first major solo exhibition in Scotland this summer as part of the Edinburgh Art Festival. At Edinburgh's Dovecot Studios, Julie Cope's Grand Tour will be on display featuring work that, in his own words, is about 'the trials, tribulations, celebrations and mistakes of an average life'. As you would expect, it's packed with detail, some of it funny, some of it poignant.

His most recent major exhibition, at London's Serpentine, modestly called The Most Popular Art Exhibition Ever!, had gallery-goers busily discussing the art and issues being addressed in a way that's almost unheard of at an exhibition. 'Well, that's great to hear,' says Perry in his flat, Essex tones when I tell him how surprised I was at that level of focus on work which doesn't look especially serious at first glance. 'That's what it's for. You want people to engage with it. The more people who are interested in art, the better it is for everyone. And I'm obviously interested in the idea of the relationship between culture and politics.'

Grayson Perry, Julie and Dave, 2015. Wool, cotton, acrylic, polyester and silk tapestry © Grayson Perry. Courtesy the artist and Victoria Miro, London/Venice

His recent one-man shows – talks, debates, whatever you want to call them – dealt with Brexit and the divided nation it has brought about as 'the elephant in the room'; in 2017 he did a TV documentary asking Leavers and Remainers to inspire a pair of ceramic pots. Meanwhile, his co-curation of last year's Summer Exhibition at the Royal Academy got five-star reviews pretty much across the board, with The Guardian deeming Grayson Perry to have saved that annual event from its 'stultified self'.

'I'm a big believer in the power of spontaneous communication,' he goes on when asked how he puts together his exhibitions and one-man shows. His 'specialist subject' is the 'interesting demographic schisms' in our country, which is maybe how he manages to be one of the few public figures able to straddle that divide. 'My absolute joy is when someone comes up to me in the street, and they're not who I imagine. I've had Rasta taxi drivers shouting out of their cars going, "I loved that thing you did!" and I'm, "yay! I've got reach!"'

Some of that reach has come from the incredibly sensitive documentaries he's done on subjects as diverse as births, marriages and deaths, and how we deal with them in different cultures (Rites of Passage) while his mini-series on masculinity (All Man) brought him into contact with people from various backgrounds, some on the scary side … 'We went into a police station and I was interviewing people who'd just been arrested, young guys caught with knives and drugs. I was knocking on cell doors going, "hello, would you like to be interviewed for a Channel 4 programme about masculinity?"' And he laughs that cackle that's become part of his national treasure status.

A Perfect Match, Grayson Perry, 2015. Crafts Council Collection: 2016.18. Purchase supported by Art Fund (with a contribution from The Wolfson Foundation) and a donation from Maylis and James Grand. Courtesy the Artist, Paragon Press, and Victoria Miro, London. © Grayson Perry

It's quite a broad spectrum of creative activity, I mention, the ceramics, the tapestries, the books, the TV, the one-man show? 'I like the variety,' he says. 'I do want to make things and I like the collaborative side of some of what I do. And then when I'm making art, I get in the zone and enjoy that. I'm very lucky to have that balance.'

But perhaps the reason Perry is best known even to the person on the street who might not have heard of the Turner Prize, let alone be able to come up with any of its winners, is the fact that he has always cross-dressed, worn make-up, hair and clothes associated with women (those clothes are often designed for him by his students at Central Saint Martins). Yes, it's a fetish thing for him; yes, it created problems in his younger life; but no, it doesn't need to be taken entirely seriously.

'When I won the Turner Prize and I was on the telly, a lot of transvestites' shoulders sagged that I wasn't the right kind of pervert,' he laughs. 'Like, "I don't want to be associated with him!" I think the average transvestite wants to pass as a woman but I got bored with that.'

So, what are the differences between Grayson and Claire? What are the intricacies of how those two personalities meet and diverge? And how does it chime with his being married and having a daughter? 'I hate to break it to you,' he says with a cackle, sitting dangerously loose-legged in that cerise outfit opposite me right now. 'But it's just me in a dress.'

And he gets up. 'I'm just going for a manly wee,' he says as he clumps across the studio towards the toilet cubicle in those cerise platforms.

Grayson Perry: Julie Cope's Grand Tour, Dovecot Studios, Infirmary Street, 25 Jul–2 Nov, Mon-Sun, 10am–5.30pm (Jul & Aug), Mon–Sat, 10am–5.30pm (Sep–Nov), £9.