Back in the '40s all cars were made the way trucks and some larger SUVs are today: in a "body on frame" style. An "H" or ladder shape made out of steel tubing that the wheels, engine, and car cabin/body are all attached to. The body as well would be made out of steel. Of note, most cars back then were designed to be rigid with strong frames and bodies that could take abuse pretty well. In a crash the body and frame of the vehicle wouldn't deform much (except at higher speeds), which meant that the occupants of the vehicle would end up absorbing much of the impact energy, with deleterious health effects (especially without widespread seatbelt use, let alone airbags).

Starting in the 1970s unibody construction for passenger cars started to become much more popular. Instead of a frame or chassis with a separate body/cabin the body itself (which was mostly stamped sheet metal) was redesigned to increase its strength in key areas, replacing the need for a frame underneath. This was one of the enabling technologies for efficient, compact passenger cars in the 1980s. Unibody construction made them lighter, translating into fuel economy and manufacturing cost savings. Additionally, starting around the 1960s car safety standards started to become more heavily regulated (with mandatory car impact tests and safety ratings from NHTSA, for example), as well as more highly demanded by consumers. Drivers didn't just want a muscle car to show off with, or a big hunk of gleaming steel to leave in the driveway as a status symbol, they wanted vehicles that were practical, cost-effective, reliable, and safe. Cars started to get features like crumple zones, shoulder belts, and later airbags. These innovations meant that the car would absorb a lot of the impact energy in a crash, leaving it destroyed but protecting those inside.

The irony here is that the DeLorean was not a lightweight vehicle for its time, it was actually pretty beefy (which is partly why it isn't as sporty as would be hoped from looking at the design). Indeed, the DMC 12 actually weighs in at within 10% the weight of a '46 Ford, so the perception of the Ford as a big bruiser and the DeLorean as a delicate, lightweight vehicle is a bit misplaced. Additionally, the DeLorean was built using a frame on chassis design, similar to the Lotus Elan. That said, it was also designed to maximize driver safety, with crumple zones around a more rigid passenger cabin.

In a minor fender bender between a DeLorean and a '46 Ford it's very likely that the Ford would receive perhaps only cosmetic damage while the DeLorean might end up with crumple zones that have done their duty. In a more serious collision the Ford would likely be comparatively less damaged, though with similar vehicle weights to each other it could sustain some significant damage. And though the DeLorean would end up out of service regardless, the story for the passengers would be the opposite of what it was for their vehicles, the passengers in the Ford would fare much worse and experience much more severe injuries.

In context, it seems as though Doc. Brown is talking about a minor fender bender, or something like a "PIT" maneuver, and on the whole it seems like his sentiment is somewhat accurate though a bit overblown. The expectation that a lightweight sports car using 1980s construction styles would be no match for a '46 Ford built like a truck and weighing the better part of two tons is reasonable. The only problem is that the DeLorean was not such a car, it was a bruiser, it had a strong chassis (even though it also had crumple zones) and in a low speed contact it probably would have fared ok, and even been driveable.

(As an extra geeky aside, earlier in the movie Doc. Brown mentions that one of the reasons he used the DeLorean was that its stainless steel body was useful for some purpose with the time machine. Given that, it might just be that he's overly protective of the vehicle's structure.)

In any event, the sentiment is mostly true though somewhat overblown, but not because the DMC-12 was lightly built or fragile compared to the '46 Ford, rather instead because it was intentionally built to handle collisions at highway speeds better.