'She lives on the same floor as me' is fine.
A strict grammarian might tell you that 'She lives on the same floor as I' or 'She lives on the same floor as I do' is more correct, but we don't tend to say either of those. These days we would all say exactly what you said : 'She lives on the same floor as me.'
You would only use 'mine' if there were a possessive construction involved, for example, 'Her address is the same as mine.'