I did not watch the mentioned video so I am not sure if what I am about to mention is discussed there or not. Also, sorry for the really long reply!

I am not aware of any available truly privacy respecting, modern cars. However, assuming theat you obtain one or you can do things like physically disconnect/remove all wireless connectivity from the car to make it as private/secure as possible, there still is little you can do to be truly anonymous.

Your car likely has a VIN and license plate as well as a vehicle registration. Assuming you legally obtained the vehicle and did not take any preventative measures prior to purchasing the car, those pieces of information will be tied back to you and your home address (or at least someone closely connected to you). You would need to initially obtain the vehicle via a compsy/LLC/partnership/etc. as the owner/renter/leasee of the vehicle and an address not associated to you. Additionally, you would need to find some means of avoiding or limiting the additional information connected to you that is likely required to obtain the vehicle like car insurance and your drivers license.

Additionally, any work that certain mechanics perform may be shared (either directly or indirectly) with data brokers - even just routine maintenance like an oil change or alignment. Hopefully you didn't use your credit card, loyalty rewards program, etc. when you had any work done!

There is also CCTV, security cameras, and other video recorders that are nearly impossible to avoid. Given enough time/resources and maybe a little bit of information, your car could be tracked from its origin to destination locations. This location history can be used to identify you as the owner (or at least driver/passenger) of the car. Unless your car never leaves your garage, you can almost guarantee that your car is on some Ring camera, street camera, etc.

Furthermore, anything special or different about your car (custom decal, unusual window tinting, funny bumper sticker, uncommon color for the car, uncommon trim/package for the car, dented bumper, fancy rims, replaced tires, specific location of toll reader placement on the windshield, something hanging from your rear mirror, etc.) all help identify your car. The make/model and year of your car can also be used to identify your car if its not a common car in the area. These identifiers can be used to help track your car via the video feeds mentioned above.

Then there are license plate readers which are only slightly easier to avoid than the video recordings. Permanent, stationary license plate readers can be found on various public roads and parking lots. There are also people who drive around with license plate readers as part of their job for insurance/repossession purposes. You may be able to use some sort of cover over your license plate(s) to hinder the ability of license plate readers to capture your plate number, but that could be used to help identify your car in video feeds/recordings.