A home mortgage and an auto loan aren’t generally structured as a single debt product. So, to answer your question, no they can not be financed together.
There are a few reasons why banks insist on keeping these products separate. First is the easiest and most obvious: time. Home loan products are engineered to span 25 and 30 years. Usually, payments on the principal of these loans don’t begin until year 15. A wide host of revenue streams for the bank are engineered into a home mortgage product: prepayment penalties, late fees and penalties, of course, interest fees, etc.
These aren’t readily compatible with the aging of an auto loan. An auto loan is usually issued over a much shorter course: ranging from 1 to 5 years. The standard revenue streams for the auto loan are similar to the home loan product: interest fees, late fees, though banks don’t typically penalize for early payments.
Though no single cohesive product is marketed by a bank that couples the home mortgage with the car loan, banks can very well issue a personal loan of specific terms to the guarantor. Personalized are collateralized. The usual source of collateral value for folks is the equity in their home. In less common cases, personal autos are used as collateral. And less common still is the option of collateralizing a uniquely valuable asset, such as fine art. Collateral is always appraised by the bank’s own appraiser. Afterward, terms of the personal loan are drafted and negotiations and bargaining begin.
A relevant product is a refinance mortgage or a home equity line of credit (HELOC), which you can use to purchase an auto from the equity in your home.
In fact, before the financial and real estate crises of recent years, it wasn’t uncommon for bankers to unofficially incentivize home mortgages with a presentation of the extent of credit that can be issued on the home equity acquired through the process of purchasing a home with their mortgage. A banker would pitch the home mortgage and in the same breath, run the credit scores and other supporting mortgage documents through a hypothetical line of credit on the equity, i.e. the borrower’s down payment.