I think this is an interesting question by the OP. I'm surprised that nobody has really tried to answer. Let me have a go at a few at least. Obviously I have varying levels of familiarity with these different brands, so I might not always be right:
Park Hyatt: A distinctive Japanese/ Danish influenced style (combined with local style), very white rooms. Also distinctive service style - A combination of some formality but without being stuffy. Perfect for the Millennial generation in my view
Aman: Known for high levels of customer intimacy, with no need to sign for cheques or to ever identify yourself, as well as the prevalence of Butlers everywhere who can handle all a guest's needs. A big focus on excursions/ events/ special occasions
Peninsula: One thing they always seem to mention is their use of in-room technology. I also always think of the doorman in their whites and hats
Como: Known for their bespoke hotel designs (architecure and interiors) and for their partnerships with top end restaurants
Rocco Forte: These hotels scream design and a slightly OTT service style. To me, the hotels epitomise what it must be like to be a billionaire living in Monaco
Six Senses: Spas. And a resort feel - I often get confused with One&Only
Dorchester: I don't know - a good place for high end prostitutes?
Luxury Collection: More often than not the old "Grand" hotel in the city
St Regis: Butler service. Bloody Marys. New York Upper East side vibe - especially in Asia
Rosewood: Solid, modern design. The Four Seasons for the Millenial generation
Taj: Natural home of the Upper Middle Class in whatever location the hotel is based