Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
Non Islamic banks do not usually purchase car to sell it to a customer rather they gives loans and take interest on the loans. The questioner's claim that when the bank pays the cost of the car means it owns the car is not correct. Owning a car means taking its custody after paying its cost and bearing all the liabilities before selling the car to a client which is not practiced by non-islamic banks.
If it is assumed that a non-Islamic bank follows the above procedure then purchasing from that bank is allowed, even though it is likeable to avoid dealing with such non-islamic institutions to avoid any possible involvement in haram activities.
Muslim scholars permitted purchasing a car from an Islamic bank since it actually owns a car and then sells it to the client. Also as we said purchasing a car from a non-islamic bank is allowed provided the dealings in this transaction-meet the Shariah requirements.
Allah knows best.