A toilet plume is the invisible to the naked eye, cloud-like dispersal of raw, infectious sewage particles as a result of flushing a toilet. Flush particles rapidly rise out of the bowl and several feet into the air after flushing. Aerosolized droplets can remain suspended in the air for tens of minutes between users, posing an inhalation risk. These invisible particles go on to coat and spread onto surfaces like floors and counters or objects like hand towels, bathmats, or even toothbrushes.
With cleaning in general, there is never fully anything. It's about abatement. If a measure is (for example) 80% effective, the whole chain forward will have contamination numbers 80% lower. So even if you keep your toothbrush in a case, it will have (again, made up numbers here) 80% less contamination. Which, depending on the microbe, it could mean the difference between a immuno-compromised person catching or not a disease.
Now, in your house, you do you. But it baffles me that the norm for public restrooms in the US, even in hospitals, is to have no lid at all.