The bumper is attached with mechanical devices. On my 1982 240D the rear bumper is an aluminum casting with some mechanical mounting clips cast into the inner surface of the bumper. These clips fit the heads of the bumper shocks, kind of like a "T-slot" arrangement. The mounting sequence is to put the bumper shocks into the clips, then run the shock tubes into the recesses in the frame, and bolt up the flanges. Once the tubes of the shocks are in the recesses, the bumper cannot disengage the clips. Older cars have a different clip arrangement, which I believe involves bolting the clip hardware to the bumper.
When the bumper gets old and the aluminum gives way to corrosion (the inside of the bumper is a place where no one usually washes, and sand with road salt collects by the pound there, leading to this problem. On the driver's side, the hot exhaust usually accelerates the process.) the clips begin to rot away. Once they are gone far enough to let the bumper begin to fall off, the solution is to either buy a used bumper from a car down south, or fabricate a new clip arrangement and bolt it into position. You may be able to convert to the older style clips, but I am not sure about that. I know I fabbed some new clips and bolted right through the bumper. This lasted for another 5 years or more, but eventually the aluminum began to crack at the bolt holes. My daughter replaced the bumper with one from another junked car at the college auto club (a W123 Diesel fanatic there donated it to her cause as the school told him to have the car towed to a junk yard or else...) a year or so ago, and that bumper is working fine. I think it has the old style clips, but I have not checked.
Good luck, and I hope this helps. Jim
1986 Euro 190E 2.3-16 (291,000 miles),
1998 E300D TurboDiesel, 231,000 miles -purchased with 45,000,
1988 300E 5-speed 252,000 miles,
1983 240D 4-speed, purchased w/136,000, now with 222,000 miles.
2009 ML320CDI Bluetec, 89,000 miles
1971 220D (250,000 miles plus, sold to father-in-law),
1975 240D (245,000 miles - died of body rot),
1991 350SD (176,560 miles, weakest Benz I have owned),
1999 C230 Sport (45,400 miles),
1982 240D (321,000 miles, put to sleep)