By this, I do not mean someone aware of how other people see them. I want to describe someone who is concerned with their personality being consistent. For example, someone who feels the need to always seem cool in front of his/her friends. Is there a word for this?
First thought was image-conscious.
Adjective: concerned about the way one comes across to other people and the impression one creates.
Somone who takes a curious amount of time to get ready; they will take the time to try and perfect every physical flaw with their image Verb
Primp (third-person singular simple present primps, present participle primping, simple past and past participle primped)
To spend time improving one's appearance (often in front of a mirror)
This person might be (or wants to be) a conformist.
1 A person who conforms to accepted behaviour or established practices
‘The final argument used by the public school is one of socialization and their failed attempts to convince the masses that children need indoctrination and must become social conformists by association with their peers.’
Adjusting your question: I want to describe someone who is concerned with their personality being congruent. Someone who feels compelled to always be congruent in front of his/her friends.
(Googled..) Definitions of congruent adjective (1) in agreement or harmony.
institutional and departmental objectives are largely congruent
(2) (of figures) identical in form; coinciding exactly when superimposed.
Use just one cut - or draw a line - and divide this white shape into two identical, congruent , parts.