Although the Estada Novo considered all of the Portuguese colonies to be part of Portugal, didn't they consider Cabo Verde (and perhaps Sao Tome) to part of Portugal proper, and Portuguese as opposed to African? Why then, would they give independence to Cabo Verde under the auspice of decolonization, yet not give independence to the Azores or the Madeiras which are (at least to my understanding) relatively similar in their economies and societies? Thanks for any responses!
Those two archipelagos were not inhabited when the portuguese landed: they were colonized by portuguese (and other europeans, but to a lesser extent), and as such, the only thing they differed from the continental people is that they were surrounded by water on all sides. This was the same with Cabo Verde, however, contrary to Azores or Madeira, they were mainly inhabited by slaves brought from other parts of Africa.
With this in mind, there was no reason for Azores and Madeira to become independent (they didn't want anyway), since they were portuguese. Still, if it interests you, there were protest in the government regarding CV's independence (some suggested they should've become an autonomous region, like Madeira and Azores are now).
the Azores or the Madeiras which are (at least to my understanding) relatively similar in their...societies
Are they though? The ethnic makeup of Cape Verde is very different to the Azores and Madeira.