Replacing a vintage 2 prong outlet that has 4 sockets?
That appears to be an add-on tap, NOT the actual receptacle installed as part of the home electrical system. If you remove the screw in the middle, the device should come off. It is likely an older version of something like this 6-outlet wall plug adapter, except only 4 receptacles and not grounded. You will likely find an ordinary duplex receptacle underneath, missing only the usual cover plate, which can be easily replaced.

The device you currently have is a bit beat up and is not grounded. I highly recommend:


If the receptacles underneath are not grounded, replace them with a duplex grounded receptacle. This is easy if they are in a metal box that is grounded, or if there is a ground wire in the cable.
If it is not possible to install a properly grounded outlet, then installing a GFCI would make a lot of sense, as that provides similar protection (others can elaborate on how GFCI is better, different, etc.) but if you need more than two receptacles (i.e., if you plan to install a new device similar to the old one) then you would need to install the GFCI upstream (e.g., at an earlier location, possibly in the breaker panel by replacing the existing circuit breaker).
If you want to install a new device to replace the existing device, make sure it is UL-listed.