An extremely self-conscious person (6)
I believe the answer is:
egoist
'an extremely self-conscious person' is the definition.
The answer and definition can be both people as well as being singular nouns.
Perhaps there's a link between them I don't understand?
'self-conscious person' is the wordplay.
'self' becomes 'ego' (synonyms).
'conscious person' becomes 'ist' (I can't explain this - if you can you should give a lot more credence to this answer).
'ego'+'ist'='EGOIST'
Can you help me to learn more?
(Other definitions for egoist that I've seen before include "Self-seeker" , "ill with him to think of others" , "Narcissist" , "It goes around with conceited fellow" , "A seld-centred, conceited person" .)