Can the metal plates in an electrochemical cell be made of the same metal?

Historically, no, each metal has its own redox potential and it is the difference between them that causes the flow of electrons with the electrolyte carrying the charge in the cell and the circuit between the metals that carries them outside the cell.

However, there are now cells that do not use the metals to create the electrochemical potentials where the metals are only conductors wherein the metals are where the electrochemical reactions take place, but are not reactants. These have to have a membrane between the two reaction compartments to keep the two reactants from reacting in solution or in the gaseous state. These are more often referred to as batteries in my opinion, or as fuel cells.