If you said "They both play a fingerpicking style with rings on their middle fingers" you would be correct.
Annie Clark is the only example of a guitar-player's guitar player in modern alt rock I can think of. I am not a fan of the Lower East-Side noise-rock tone she goes for. She tends to play in a very staccato style. But she's really quite a fine player.
But there are a couple other things she does in this video which I want to steal. At 8 minutes there a nice funk riff which might could use some interesting variations applied to it. At 10 minutes there's an interesting harp-like thing which totally sounds like Pleasure for the Empire to me.
§They're also both from Texas. Could be a coincidence.