No, Sipser's textbook does not cover Chomsky hierarchy, a containment hierarchy of different classes of formal grammars. Nor does it cover that hierarchy with a different name. In fact, the book does not mention any formal grammar other than context-free grammar.
That textbook does cover many related concepts such as regular language and finite state automaton, context-free grammar and context-free language and non-deterministic pushdown automaton, linear-bounded deterministic Turing machine, recursively enumerable and Turing machine. It certainly covers context-free grammar pretty well. It is not surprising the post linked in the question recommends it for understanding formal grammars.