Is it actually true that boiling water will disinfect a toothbrush?

Yes: It will likely need to be in the boiling water for some time though. I would think this would damage a lot of toothbrushes too.

It can, but....: First, there is a difference between sterilizing & disinfecting something. Sterilizing kills all organisms. Disinfecting only kills some. Placing a tooth brush in boiling water will disinfect or sterilize it depending upon how clean it was to start & how long you boiled it for. However, it will ruin the bristles & make it substantially less effective. You can rinse it in an antimicrobial rinse.