Should 2 root-canal tooths and a good permanent tooth be fused together like one tooth?

What your dentist has done for you is called "splinting".  This technique of joining multiple crowns together is most frequently done with periodontally involved teeth that are mobile (somewhat loose) because the periodontal disease has destroyed much of the bone support.

Root canaled teeth are always weaker than teeth that haven't had root canals.  It appears that your dentist feels that one or more of your teeth are so weak that he wants to protect the tooth and/or root from having to sustain strong loads, possibly causing fracture and loss of the teeth.  That is a judgement call, and without seeing the teeth and x-rays I don't think any dentist could criticize him for doing this for you.  Other options would be extraction, implants, bridges or just doing single (unsplinted) crowns and taking the risk.  I'm sure you would prefer what he did than those other options...right?

There are two challenges with splinting multiple teeth together with crowns.  One is that it will be much more difficult for you to clean your teeth with dental floss, thus you may have to be diligent and use a floss threader or superfloss to make sure you remove all of the plaque and don't end up with gingivitis or periodontitis from neglect.  Your dentist should instruct you how to clean and floss around splinted crowns.

The other issue is one of "fit".  It is much more challenging for the crowns to fit well at the margins when crowns are splinted together.  If one or more doesn't fit accurately, this could result in leakage and decay that might end up going up under the crown.  Splinted crowns that don't fit properly on one or more teeth can usually be discovered by taking x-rays before (preferably, if the dentist has any doubt) or after permanently cementing the splinted crowns into place.