The client is essetnially the person that you are providing the service for directly, so the advisor will in fact be YOUR customer but it all depends on if he employs you to do the work or he is hiring you to do the work !!
I'm not sure about the AML checks but im guessing you would not need to do that so called 'legal' side of things, that would be more what his supported instituion deals with....
The fact that the advisor has his own clients/customers is irrelevant to you it just sounds like the advisor is getting you to do the work that he does not have time to do himself (but like i meantioned above you need to establish if he is employing you, like a accountancy practice or he is hiring you as a self-employed worker because Tax and NICs are differently recorded).
but not sure how the advisor can pay you and also pay the invoice on the client's behalf, that sounds like he is paying twice??