Shower Water Handle central plastic core is stripped out - how to replace?

The first thing I would try is getting a screw that's a little bigger and see if you can screw it in tight. If that piece was metal, I'd say tap a new thread but if it's plastic as you think it is, a self threading screw might do the trick.

If not, you'll have to replace the cartridge. That little central part can't be replaced, It's part of the cartridge. That outer sleeve should be removed, pull it straight out or turn it counterclockwise. Then use an adjustable wrench or a pair of Channelocs to remove the retaining ring for the cartridge and then remove the cartridge. The water to the shower has to be turned off to do this. If the shower doesn't have a shutoff behind the chrome cover, then the water to the house has to be turned off.