YES you can "lift" the car from these points and yes over time it will damage the underbody of the car, i have a '76 280z, its unibody over the years of it being taken to shops for tire changes, the floor board has been pushed up and damaged....

It blows my mind that modern shops don't have the proper equipment to lift a unibody car, i mean is there even a solution or are we supposed to use the small circular pads on the side to the jacking point

the issue here is strength, the unibody isn't designed to support the body of the car from the floorpans, what is pictured previously is nothing more than a piece of metal welded to the floorpan, frame rails on older cars are the main support for the body of the car so lifting from them wasn't an issue..


This is the jacking adapter you should use at home,

 

I have to disagree, if they left a dent they didn't lift it correctly. Damage to the underbody is not part of a repair or service. My car has been on a hoist the usual number of times and it has no missing paint or dents.

 

I've owned this car for 8 years now and never had it lifted at the frame rails so I was a bit worried. So, aside from the cosmetic damages, I guess everything else should be okay, right? I don't know much about the unibody frame rails and how they function on these cars.

 

From viewing the picture, it looks like the frame was flattened, not dented, where contacting the hoist. It would make me angry but this isn't that big of deal. If it was deeply dented or crushed you then would be losing strength.

The missing surface coating is more of a concern, especially if you drive in snow and salt. I'd suggest a paint repair or at least coating the area with grease.