The Clicker item seems to change the character to look/behave like another, but does this mean that all the achievments are going to account for the new character?
As with all items that change which character you're playing as (for example, Judas' Shadow, Lazarus' Rags, or the Ankh which turns you into ??? (Blue Baby)), finishing a run with that "new" character will unlock achievements for the "new" character only. The clicker changes which character you're playing as in the same way, so you will only unlock achievements tied to whatever character you are using at the time the achievement unlocks.