To unlock the Feature Car mode you will need to first purchase the Featured Car, this can be seen in the dealership and will have a trophy symbol which you can spot in the screenshot below.
Remember that you will need to be in the right Tier for the Feature Car, for example if you are in Tier 2 and have purchased the Jaguar E-Type, you will not be able to play the Feature Car mode as you will not have unlocked the car's Tier.