One of us feels like having things joint or combined is just normal as to how a marriage is- and the other feels combining bank accounts, car ownership, etc. is a loss of individual identity and that these types of items should be bought separately and owned individually as one or the other's property.
Well, by law, it doesn't matter whose name something is in, if it was bought after the marriage, it belongs to both of you. And in some places, the home you live in belongs to both now even if the paperwork still only has the name of the one who had it before the wedding.
If 'your' car is the brand new expensive sportser and 'her' car is the old run-down family wagon, you're still dividing them both up in the event of divorce, no matter what name(s) are on them.
There are two views of marriage these days. There's the traditional merging of finances, becoming one financial entity, no matter who earns the money. And then there's the financial partnership, where both remain individuals and keep some things separate.
If you marry young and establish yourselves after marriage, and stay together a long time, you're more likely to be the first type. If you marry later in life, either because of establishing a career first, or it not being your first marriage, I think you are more likely to be the second type. Especially if there has been any sort of financial infidelity in your past!
So I'm thinking that one of you has been burned in the past and wants to maintain some independence as security.
Honestly though, calling something 'your' car or 'my' car is done no matter what's on paper or how finances are handled. It's the car your partner drives vs the one you drive. Computers and furniture may be bought out of joint funds, but still get called 'your' computer or 'my' side of the bed for ease of reference.