There's no doubt that, between you and your wife, you own the vehicle. Look back at last years tax return and see what your vehicle deduction was. You probably need to amend the return if you said that you didn't own it.
If you do not own or lease the vehicle you can't use the standard
mileage rate. That's because the standard mileage rate includes an
allowance for depreciation (wear and tear) and only the owner or lessor
is allowed to claim that.
Also, if you don't own the vehicle, your actual expenses will be under-reported, missing the depreciation.
Of course, this year you should say that you own it. Note if there's is a noticeable difference in deductions, all things being equal.