From now on, we are going to use the symbol a to refer to the same number that b refers to.Although it's a little silly to be using two different symbols (a and b) to refer to the same number, there's nothing mathematically wrong with it. The letters used in algebra have no special pre-defined meaning; they're just shorthands which can be used to represent anything that one wants to come back and refer to again.
Would there ever be a situation in which saying "let a=b" would be useful and not just silly the way it is here? Yes indeed. For example, you might be talking about two not-necessarily-related things a and b, and in one part of the argument you might want to think about what happens in the particular case that they happen to be the same thing. You could use the phrase "let a=b" to introduce that part of the argument.