broken rear bracket
Hello! What you have here is the old Merillat drawer socket. These aren't made anymore but our 32-119 can be used as a substitute, as long as you're okay with drilling new holes.
As for removing the original, these were always just plugged in using little nubs. They should just plug off, but if you're getting a lot of resistance then that means the original installers might have actually glued them in place. That's no good.
The first thing you should try doing is removing the bracket with a putty knife. Just slip it behind it and pop it off. Just be careful not to force it; if it is glued in then you risk damaging the back wall.
If that doesn't work, try a little WD-40, let it sit for a while, then try prying it loose again. If it's still not coming off then you should consider purchasing a glue removal solvent from a local hardware store.