Why does a touched piece not have to move in online play? There is a rule of playing the game of chess that a touched piece of one's own should be the one to move. I have always been taught by professional trainers from early childhood that this is the official rule to be followed in tournaments, but not necessarily in friendly games.

However, I find that online interfaces for playing rated games do not enforce this rule. I often find myself starting to make a move in blitz and then opting for another, possibly with a different piece. The interface always allows that in all the sites I have played.

Is this an omission on the part of the designers or rather an indication that chess players really do not care about this rule (mindlessly repeated from generation to generation)?
I think the reason becomes more apparent when you consider why the rule is in place for OTB games - an opponent constantly moving their hands around the board and moving the pieces around can be very distracting while the other player is likely still trying to concentrate on the position. 

By contrast, in online chess both players are using their own screens and the fact that one player starts moving one piece, decides against it and then moves another is not even conveyed to the other player - they see a static board until the final move is made.

Another factor (but not the main motivation IMO) might be that, as you hinted at in your question, most online games are seen as 'friendly' games - it has only been very recently that the idea of officially rated online games has been accepted by FIDE, for example.