The metals used are the same base element to avoid the electrolyte and the electrode reacting with each other and inhibiting the function of the cell.
Cells with different electrolytes and electrodes do exist, such as a specific plating cell I forget the name of.
However, if a cell has a zinc electrode and a copper electrolyte (assuming these do not react and will plate a key sitting in the electrolyte) the key will be plated with copper for a time as the copper comes out of solution. However, soon there will be more zinc coming into solution than there is copper in solution. Due to this the zinc will be plated on top of the copper which is on top of the key. (I am not sure if this arrangement would actually work).
This technique is used in cells such as jewellery plating cells to get coatings of two different types of metals on the jewellery.
Essentially, the same metal is used for the electrode and electrolyte to maintain consistency across the plating and solution.