This was a common technique. It's fallen out of favour in more recent times due to the introduction of active power factor correction. If you see a power supply with a simple slide switch to select between 120V and 240V then it's probably using this technique (on the other hand, if you see a device using a rotary switch or a movable insert to select voltage, then it's more likely changing taps/winding configuration on a transformer).
In this configuration, two of the diodes take all the current, while two are always inactive, so it might be a good idea to replace them with beefier versions or rewire the two redundant diodes so they are in parallel with the active diodes.
Also if it is desired to maintain the same level of ripple the original rectifier had, larger smoothing capacitors will be needed.
Obviously this power supply does not provide isolation, and the "common mode" voltage on the DC side will depend on what is going on on the AC side, but that applied before it was modified too.
One way this does differ from the behaviour of a bridge is it's asymettery. With a bridge, the common mode voltage on the output would be affected by both poles of the supply. However with this design, the common mode voltage of the output is mostly driven by the pole of the input that is linked through to the output capacitors. If the capacitor center tap is tied to the hot conductor then the common mode voltage will vary through the cycle, if it is tied to the neutral, then it will be mostly stable through the cycle.
Still that isn't hugely relevant in practice. In practice, both the hot and neutral connections from the mains should be treated as dangerous and hence so should the output of the rectifier.