You can’t give away someone else’s property without appropriate legal authority such as a power of attorney. Even then, there’s a requirement on the attorney to avoid any transaction which is not in the best interests of the donor.
If the car is registered in both names, both parties need to sign the transfer.
If the car is not registered in both names, it gets a little more complex. Being the registered owner does not guarantee ownership - they print a statement as much on the certificate of registration (or at least they used to, I haven’t checked ina while). If the car is transferred without one owner’s knowledge, their interest can be reclaimed if the vehicle wasn’t transferred for value (eg gift) and/or the transferee is aware of the person’s interest in the vehicle. This gets disrupted if the vehicle is further dealt with in some way, such as on-selling or granting a third party a security interest.
If there are sufficient acts to disrupt the interest, then the wife has a cause of action against the original co-owner.