So theres a cartridge Bottom bracket unit installed in the bike frame, and the spacer on the left side has sheared leaving some-amount of the lip screwed in and working, but not easily removable.
And the cartridge is not fully seated on the right hand/drive side.
If the cartridge is in the frame enough to give you a good chainline from the chainrings to the cogs at the rear, then you're probably okay to ride. The internals of the cartridge do the real work, and the left/NDS is under less load than the right hand side but it still has to resist/transmit the left leg's input.
Downside of this approach is that in a few years, it will be extra-hard to remove the plastic remains, and they won't have as much body/thickness/meat to resist squashing loads and will deform some amount over time. This will mean the threads on the righthand/drive side are under cyclical stresses and may get damaged.
If the chainline is off, you'll have accelerated chain wear, and increased chain drops probably to the inside. In this case, I'd unthread the cartridge, and then try unthread the plastic stub until you can get a good chainline.
If the remaining plastic is damaged, I'd weaken it with a fine hacksaw and then use a cold chisel, punch, or a flathead screwdriver to snap it out. Take the cartridge and stubs of plastic into your preferred LBS and ask if they have an aluminium spacer - many good LBSs have junk like that in the spares drawer. If not, buy another new cartridge with an aluminium spacer, and remember to save it - they rarely wear out.
Personally, I'd get a new spacer of aluminium, even if it means buying another cartridge.
I suspect the plastic support snapped because of one or more from:
- The thread in the BB housing had some metal obstruction which stopped the threads from engaging smoothly.
- A casting defect in the plastic had the same effect (unlikely, Shimano's pretty good)
- Some grit was in the thread (it may have felt crunchy while threading in) Could have been reduced by cleaning and greasing the threads before assembly.
- The cartridge was somehow off-axis and the spacer was tensioning against the housing.
- The threads on your frame are deep, and the spacer was bottomed out on the lip, but you kept tightening.
- You were powering it in with a drill driver or worse, an impact gun/rattlegun and had no sensation/feedback of the resistance.
Figuring out how it happened is key to not doing it again.
I've had something similar happen, but for me it was the drive side of a cup/cone and it was worn out. The outside of the cup dropped, all the balls scattered down the road. Ended up using a hacksaw to carefully remove the hardened steel piece, and fitted a BB cartridge after cleaning up the threads and swarf.