Tables of $\mathbb{Z}/6\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$? 
Compare the multiplication tables of: $\mathbb{Z}/6\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$:

For $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$

For $\mathbb{Z}/6\mathbb{Z}$:

I also have found a proof that they are isomorphic to each other and that they both have order 6.
I don't know if this interpretation is correct or enough.
 A: Group Cayley tables satisfy the Latin square property. Your tables do not. The reason: you used multiplication instead of addition.
A: Every group has an operation. For some groups this operation also has a 'real world' interpretation (like addition, or composition of functions, or actual multiplication of numbers) but the charm of group theory is that by zooming out all these seemingly different real world operations are examples of the same thing: a group operation.
Now some people will call the operation in any abstract group 'multiplication' regardless of if that matches the real world application or not. The person who gave you this assignment is one of those: the 'multiplication table' should better be called 'operation table'.
You have to forget about what else you know about these groups besides them being groups and just make the table for the group operation.
What Shaun tries to tell you in the other answer (and comments) is that in these cases the group operation is what in the real world would be called addition. So in these particular cases the table we want is the 'addition table'.
But Shaun and I know that that is what is wanted for two reasons: calling such a table a 'multiplication table' is quite common (even if Cayley table is a better because more neutral word) and 2) addition is the only 'natural', 'pre-existing' operation on these sets that makes them into a group. It is true that there is multiplication too, but looking at these sets with multiplication as a the operation wont make them into a group.
(It is a nice, but unrelated, exercise to check what part of the definition of group is violated when using multiplication as the operation.)
