I think it's Fudge's bowler.
Reasoning:
I follow my master wherever I go: Fudge always has the hat with him
What he's thinking, I might know: It goes on his head, so it might know what he's thinking.
I'm always close to his mind: Again, on his head.
I'm unique and one-of-a-kind: It's the only green bowler hat ever mentioned.
My skin might be green: It's green.
But I do not mind being seen: It's showy and Fudge always has it. Harry always notes it when he sees Fudge.
I am occasionally taken for a spin: Not sure on this line, but maybe Fudge plays with it or it spins when he goes through the Floo network or whatnot.
So let the riddle solving begin
I could be wrong but it all fits so well.