Obviously there were plenty of great groups and artists in the British invasion of the 1960s, and in many ways this movement remains the gold standard of rock music in general. But most of them were more American than Americans, e.g., early Stones, Animals, Yardbirds, Small Faces, Dusty Springfield, Them, Hollies, all of whom were slavish devotees of Cosmo-American music.
The Kinks, however, were different. You will notice, for example, that most of the great British groups didn't sing with a British accent. In fact, one of the reasons the Beatles were such effective rockers is that the Liverpool accent is closer to American. In particular, they pronounce a hard "a"; imagine how different it sound if it were I Want to Hold Your Hond or It's Been a Hod Days Night, or Kant Buy Me Love. It's difficult to sing rock with a lilt in your voice.
Peter Noone also sang with a conspicuous British accent, but Herman's Hermits don't quite make the cosmic cut -- although they are definitely underrated, or unfairly maligned. They especially served a niche that opened up when the Beatles started producing more serious music in 1965, with Rubber Soul, and then 1966, with Revolver. The Monkees then rushed in to provide that service in late 1966, ousting the Hermits from the stage. Teen idolatry is a vicious and unforgiving idiom.
Another British group that definitely sounds British is the Zombies. In fact, they threw some American soul and R & B into their sets, which always sounded a little silly. Imagine Harry Potter singing I'm a Road Runner, Honey!, or I Just Want to Make Love to You.
The Who had some British elements and concerns, for example, Quadrophenia, the subject of which -- Mods vs. Rockers -- was a wholly British phenomenon. They were also fairly ineffective at trying to reproduce straight covers of American R & B, and didn't come into their own until they developed a unique style of bombastic order from chaos. But that style was greatly influenced by the emergence of the heavier sound of Hendrix in early '67.
David Bowie was pretty British, especially early on, in that he retained some Kink-like elements of British "music hall" style (e.g. Hunky Dory). I would also nominate Fairport Convention, who were to British folk what the Byrds were to American folk; and Pentangle, who also threw in some jazzy elements, forging a unique blend of British folk-jazz.
I first became a big-time Kinks fan with the release of the Kink Kronicles in 1972. They had hardly been heard from in America during their golden age between 1966 and 1972, so the album came as a revelation. The music they produced during that time didn't sound anything like their contemporaries, which is one reason why it still sounds fresh. Like most great composers, Ray Davies created his own unique musical world which is at a right angle, so to speak, to profane time.
Kompulsive Kinks kollectors are all akwiver about this upcoming six-disc box set of BBC performances, from the mid-'60s all the way to the 1990s. (This is without question the best existing Kinks kompilation.)
I stumbled on the following video this morning. It's touching where Davies gets choked up at the beginning of Days, since the performance was dedicated to his recently departed bandmate, Pete Quaife, the original Kinks bassist: