The lines seem to all be references to
Their songs (Do I wanna know, R U Mine, etc)
I'm a puppet with no strings
"I'm a puppet on a string" - R U Mine
And a modern lover might wear my namesake with her interstellar apparel
"Arabella's got some interstellar-gator skin boots", "But she's a modern lover" - Arabella
I tell you that I do wanna know
Do I wanna know
What you would call me no matter your altitude
Why'd you only call me when your high?
I'm not the kind that goes up to your knees
Almost everyone has wanted me to be theirs
I wanna be yours
R U ready to guess?
R U Mine - I sure hope so!
Sock Monkey - type of monkey (like arctic monkeys), wear them with their nightgown, This sock doesn't go to their knees, sock puppets have no strings