Dress down (mostly)
New York has a reputation for being excessively fashionable, but in reality, overdressing comes across as trying too hard, while underdressing is, generally speaking, totally cool. For instance:
You can wear sweats at Bergdorf’s, and bring a dog
Bergdorf-Goodman is one of the world’s tonyest department stores, but its sales associates are trained not to judge customers by their outfit -- why should people with money to spend on dressing up have to dress up while spending that money? -- and if you need a second opinion on that new jacket, you can bring in your very discriminating pooch.
You can even wear jeans at Ivy League clubs
After decades of snobbish resistance, denim "in good repair" is acceptable in select areas of the Yale Club, and in select areas -- or possibly the entirety -- of the Harvard Club (the terms are vague, presumably purposefully so). Both clubs welcome jackets, but do not require them. Meanwhile, the Soho House actually outlaws the wearing of suits.