Not really, since the term 'Achaean' was always taken as a catchall term for all Greeks. Achaea was most likely (one of) the names by which Greece was known. The Ahhiyawa correspondence in the Hittite texts supports this further.

So it's not a similar situation to, say, the Spartans being celebrated for their role in holding off the Persians. Any Classical-era Greek reading the Iliad would understand himself and his ancestors to be included as Achaeans.