Choice of Lewis Acid for EAS reaction
Honestly speaking I think there won't be a difference as in clayden they have used $\ce{AlCl3}$ for bromination


If I had to give a reason it would be that when the $\ce{ AlCl3-Br2}$ complex is made there might be a teeny tiny chance that instead of bromine breaking away the chlorine  breaks away. 



My basis of saying that the probability of chlorine breaking away is low is because there's a +tive charge on $\ce{Br}$ in the complex.Chlorine on the other hand is free from any such effects. There are no factors supporting the breakage of the $\ce{Al-Cl}$ bond. So the chances of $\ce{Br-Br}$ bond breaking are higher than the $\ce{Al-Cl}$ bond. 

The same reasoning can be applied for $\ce{AlBr3}$ with $\ce{Cl2}$

That's why in Master Organic they've written preferred not necessary.