More culture shock.
In Shakespeare-era theater, women were all played by men. But for well over a century, women are almost always played by women, and that carried over into movies and TV shows as they were developed.
Men's clothing would look fairly familiar, but women's clothing would be startling. Women typically wore ankle-length skirts and dresses until about a century ago, when knee-length ones started becoming common. Knee-length is still the most common length, though shorter dresses and skirts were prominent half a century ago. An even bigger change also occurred half a century ago, when many women started wearing pants/trousers. Nowadays, in many places, a large fraction of women do that.
A sixteenth century Englishman likely wouldn't have ever encountered trousers. And he would be astonished by the disappearance of fashion and flamboyance in men's clothing.
In the C16th, a man bought clothes the way C20th men bought cars - they were an expensive investment, and few people of means would settle for a suit of clothes that just did the job of keeping him warm and modest - clothes were a statement, and just as a wealthy man today might buy a supercar, to signal his wealth and power, so C16th men bought clothes to show off their wealth and status.
That today the King of England wears much the same charcoal grey suit as a bank clerk would completely flummox our time traveler.
Take a look at C16th portraits of powerful men, such as Henry VIII, or his courtiers. Even the puritanical types were making wealth statements through their clothes - black cloth was very expensive, as were very white linens (with cotton only available to the very richest in Britain), and bright reds, yellows, blues, and greens were all costly. Wearing grey, brown, or un-dyed clothing was the sign of poverty, and anyone who could afford expensive dyed fabrics would wear them. There were even "Sumptuary Laws", that attempted to prevent people from wearing clothing above their station (a sure sign that many were doing so).
Even as recently as the C19th, when Victoria married Albert, his clothes for the wedding were picked over by the press just as much as hers were. The assumption that the groom will wear a black tux and the bride a white gown is a late C19th and early C20th thing.
For men to eschew displaying their calves (in colourful linen hose, and tied at the knee with a flamboyant ribbon about the leg of his breeches); and would even have discarded the codpiece as a fashion statement, would be something of a surprise. Breeches steadily got longer through the C16th, and by the C17th were tied below the knee, rather than above as in the early C16th. This likely reflected reduced prices for the woollen fabrics used in their tailoring, allowing even the middle classes to afford the larger amount of material required. Codpieces steadily declined in prominence during the C16th, perhaps as a reaction to the decreasing male virility of the court, first as Henry VIII aged and became less healthy, and then with the ascent of Elizabeth to the throne (bearing in mind that both monarchs were totally autocratic dictators, to a degree that would have made Stalin envious).
Many C16th doublets and breeches were 'slashed', to show off the colourful (and therefore expensive) linings and facings. Men in the C16th were bigger buyers of fashion than women, and they dressed to impress.
That your comments on fashion today focused mainly on women's fashion would be a surprise in itself. As would your assumption that men's fashion would not have changed much (an understandable error, as it genuinely didn't change in the last three or four generations; your great grandfather wouldn't see a vast change in men's dress; But his great grandfather assuredly would).
