It's one o' these...
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To remove it...
Grab the flange with a needle-nose pliers and bend it outward. Flex it in and out a few times and it'll pop off. Do the same for the other half if it splits.
Now just push it into the wall.
To try and salvage it...
It's possible that it simply wasn't installed properly. That type of anchor has a collapsible accordion-style tube on the back. You have to run the screw in fairly firmly to get it to do its thing. Its thing may simply not have been done.
To do that, hold the flange with a pliers and try running a suitable screw in. Once it bottoms out, go a bit further. If things are going right you'll feel an initial release, then it'll snug up again.
To replace it...
Remove it as above, then either try another type of winged or toggling anchor, or repair the drywall and start fresh a couple inches over. Don't bother with plastic plugs. They won't hold in this situation and aren't great to begin with. You want positive latching against the back of the drywall.