'angora' originates from Turkish, specifically the place name 'Angora' (now 'Ankara'), where long-haired goats and rabbits were bred; 'lined' is from the English verb 'line' (to furnish with a lining), so 'angora-lined' is a compound formed from 'angora' + 'lined'.

'angora' entered English via French 'angora' (from the Turkish place name 'Angora') and came to denote the hair/wool of animals from that region; English then formed the compound adjective 'angora-lined' by combining 'angora' with the past-participial adjective 'lined'.

Initially 'Angora' referred to the place and the animals (goats/rabbits) from that region; over time it came to mean the soft wool or fur itself, and 'angora-lined' evolved to mean 'provided with a lining of angora'.