Are the units of rate constants affected if we have more than one substance in the rate law
The heart of your problem is that you're assuming that there exist different types of the unit mole. 

Consider the unit of length one meter. Take two meters of cotton thread, and two meters of jute thread. Place them together and you get a nice square of $1\times1 \text{ &lt;unit of area&gt;}$. Now what should the unit of area be? Should it be $=\text{cotton meter} \times \text{jute meter}$?

Well, it should NOT be $\text{cotton meter} \times \text{jute meter}$ because a meter, being a unit, is same for all substances being measured. A meter of jute is exactly the same length as a meter of cotton. Hence, the unit of area should be $\text{meter}^2$, which it always is.

Similarly, in your solution, you are assuming there exist two different types of the unit mole, one for $x$ and one for $y$. However, there is only one fundamental unit, and it measures the same amount of substance for both $x$ and $y$.

Hope it helps!