Assumed name accordingly what Parisian adopted by British provides (9)
I believe the answer is:
sobriquet
'assumed name' is the definition.
(a sobriquet is a nickname)
'accordingly what parisian adopted by british provides' is the wordplay.
'accordingly' becomes 'so' (synonyms).
'what parisian' becomes 'que' ('what' in French).
'adopted by' indicates putting letters inside (I've seen this in other clues).
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'provides' becomes 'rit' (I can't justify this - if you can you should believe this answer much more).
'b'+'rit'='brit'
'que' put into 'brit' is 'briquet'.
'so'+'briquet'='SOBRIQUET'
(Other definitions for sobriquet that I've seen before include "A different name" , "Another handle" , "dubious name" , "Person's nickname" , "Byname -- Boers quit (anag)" .)