Technically... couldn't He loves my pink bike work? because... when you say của mình... it means mine. So technically you could take this sentence either way you want. He loves HIS pink bike... OR He loves MY pink bike... right? :)
I get the impression. Của mình refers to the "self" of the main subject/person in a sentence. While của tôi is the "mine" or "self" of the speaker. Example: He took his shoes = của mình ; He took my shoes = của tôi.