Alright for a Greek character to have an animal (5)
I believe the answer is:
okapi
'an animal' is the definition.
(thesaurus)
'alright for a greek character' is the wordplay.
'alright' becomes 'ok' ('OK' can be a synonym of 'alright').
'for' says to put letters next to each other.
'greek character' becomes 'pi' (letter from the Greek alphabet).
'ok'+'a'+'pi'='OKAPI'
'to have' is the link.
(Other definitions for okapi that I've seen before include "Giraffe-related mammal" , "Giraffe-like animal with stripes" , "Hoofed mammal" , "Mammal of giraffe family" , "Ruminant related to the giraffe" .)