A learner - one with a second false name (5)
I believe the answer is:
alias
'false name' is the definition.
(synonyms)
'a learner one with a second' is the wordplay.
'learner' becomes 'l' (as in a learner driver with L-plates on their car).
'one' becomes 'i' (Roman numeral).
'with' says to put letters next to each other.
'second' becomes 's' (common abbreviation).
'a'+'l'+'i'+'a'+'s'='ALIAS'
(Other definitions for alias that I've seen before include "aka" , "Handle that isn't genuine" , "stage name, perhaps" , "another name" , "False identity" .)