There is no "implied verb". "Like you and me" is a shortened way of saying "like you and like me". You can know the correct me/I choice by deleting the 'you and'. Better, you can know it by remembering that we use the pronoun me, along with other objective pronouns such as us, him, her, you, and them, when the pronoun is the object of a verb:
Just like me/him/you/them
Danny thanked them.
The dog followed John and me to the door.