Some do, some don't. I have family members who identify as habesha and oromo, others who just identify as ethiopian-oromo, etc.
The term 'habesha' is almost like the term 'black' in the west, in that, some africans identify with it, others don't. 'Habesha' usually 1 of 3 meanings:
1) Any Ethiopian. Or the more prominent groups who look alike: amara, oromo, tigrayan, gurage, etc
2) Only the Ethiopian groups who speak a semitic language, are orthodox christians, etc.
3) Some use it as a collective term to refer to both ethiopians AND eritreans.