The solution is pretty simple.
In career mode you race for money which allows you to buy new cars.
In arcade mode you have a sponsor who buys you any car you want.
The garage in career mode only shows career mode cars. The garage in arcade mode shows your career mode cars + arcade mode cars.
All tuning and customization options and driving physics are the same in career and arcade mode. From now on, only difference is how you get new cars and what is shown in your garage. You can use any car you own on any track you want, by creating your own race in career mode, or any car the game offers (thus which you paid for) in arcade mode.
When you go online, you select a car from your arcade mode garage (=career + arcade mode cars).
This would go against Kaz's obsession of not letting you enjoy your game the way you enjoy it most though. He deliberately puts obstacles such as short A-spec (=grinding), locked cars, trade limit, and instead of removing those obstacles he gives you the option to pay to overcome those obstacles. Family, passion, vision,... Yeah right, money.