Suppose you don't want anyone to know where your car comes from, who's owned it before and even where it was made. Is it possible to make this car "unidentifiable" even after thorough forensic examination? If yes, what can be done, in terms of changing plates, erasing registers, chassis, etc, to achieve this goal?
A Ford body will always be a Ford body and even the year model can be worked out from it. A BMW engine is still a BMW engine and this can be worked out from the design.
With a chopshop car, you change all the bits with parts from other cars so whilst they tell it's a Ford body, nothing else is Ford. Once the certification plates are removed and the engine number is gone, they'd be down to trying to trace parts which gets awfully difficult.
A large part of this question depends on the model of the Car
In most cases origin and ownership of a car can be obscured by erasing the VIN number which is usually placed on various parts of the car.
It is pretty much impossible to erase who manufactured the car. First of all, most models can be easily traced back to a company of origin and likely factory by appearance alone. The materials in the car can be used to identify factory of origin right down to the metals in the engine (if they cared to look that far). You could obscure this to some extent by replacing parts from different suppliers but to what end really, at some point you've become the manufacturer.
Assuming you have a factory-built car, and haven't made severe body alterations, the answer is pretty much no. As mentioned, not only will the manufacturer and model be obvious to a regular car buff, they'll know where it was made, because most factories tend to specialize. For instance, suppose your vehicle is an F-150, one of the most common pickups in North America. It's either from the Kansas City Assembly Plant or the River Rouge Complex in Dearborn.
Then, there's the vehicle VIN, typically found engraved in assorted locations on the vehicle, including places that require the removal of major components such as the engine or in the case of the F-150, the cargo bed in order to access. And in some cases it's stamped into metal such that you can't sand it down to remove it without making a hole right through the component.
Next, serial numbers on parts. Again, unless you've replaced everything, they will be recorded somewhere
Then, there's another practical issue: how do you drive it? Car registration and insurance requires a vehicle's VIN. Sure, you might go through all the effort of creating a non-traceable vehicle....but you're not going to be driving it anywhere without being pulled over by the first traffic cop you happen to drive by.
Identifiability depends not only on the car, but also on the records. If the VIN and engine number are sanded off and there's no license plates on the car, no registration card in the glovebox, how would someone match it to a record? If it has identifying dents and scratches, and you have a photo of the car you're looking for, you might be able to match it. The previous owner, if he saw it, might be able to point it out. If it has distinguishing features like bumper stickers, people from the same town as the owner might recognize it. If it has major repairs, you might be able to ask around at the repair shops in the town to see if anyone remembers doing the work. If it is stolen and the owner has recently filed a report, the police might look up the report and discover the match.
But basically, if you are far from the place where the car was obtained, and there's no recent police report specifically describing it, and if you have removed the easy forms of identification, it may be impossible to trace. Or at least practically impossible, such that no one can or will make the effort.
The one and only way to do this is to build every aspect of the car — every nut, every bolt, every stitch of fabric, every mole of glass, everything — 100% custom in your own garage. Why?
Because you can track bolts, glass, paint, parts, assemblies... even if you remove the model numbers, serial numbers, registration numbers, from every fuse, plastic part, bulb, etc., you'd always have something you can trace.
Which is a long way of saying, no, it can't be done.