For one thing, if you think Deuteronomy 22:5 still applies, this is repugnant to Scripture.
For another thing, you are not comparing two things that are equivalent. The fact is that very few women wear male clothing. What most women wear is clothing that has been adapted for women from male clothing. Women who wear pants, for example, do not typically just wear men’s pants. They wear pants that are narrower around the waist, wider around the hips, fuller around the buttocks, and less spacious around the crotch. The same with a suit coat/sport coat etc - it will be cut to fit around the woman’s bust and waist, the shoulders will be narrower, and it buttons up the other way.
Also, this is culturally specific. I will admit, my wife does not wear pants except as a utilitarian garment within our home. That is her preference, not mine, although I do not disapprove of her choice. Our daughters also have grown up wearing dresses and skirts as a norm. We did not hold a gun to their heads, but that was what was normal in our family. However, we are not oblivious to what happens in the rest of the society in which we live. We know that most American women and girls do wear pants a lot of the time. The fact is that British men do not normally wear dresses in the 21st century. Mr. Styles is being deliberately provocative. He is deliberately rejecting gender norms for the society in which he lives.
The other point, which is related to my first point, is that the dresses that Mr. Styles wears do not appear to have been reinvented as some new trend for men wearing dresses. Maybe in 100 or 200 years there will be dresses for men and it will no longer seem weird. However, if men are routinely wearing dresses in 100 or 200 years, my prediction is that they will be men’s dresses, not women’s dresses, just as women do not wear men’s pants. Dresses are designed to flatter the shape of a woman’s body (narrower shoulders, a bust, waist, hips, proportionately longer legs) and to complement a woman’s overall appearance (e.g. not having facial hair or hairy chests and arms like men). Therefore, just putting a man in a woman’s dress does not look good. And yes, I know that not all men and not all women have the same body shapes, and that is why you choose clothing that suits your particular shape, but choosing from clothing designed for the correct sex is a good place to start.