Now, why do they use "odd" instead of "handicap"?
Sometimes the term are used interchangeably. The term "Odds" in chess dates back at least to the 18th century and probaby earlier. It was commonly used throughout the 19th century, even though late in 1800s you can read about handicap tournaments. Int the 20th century writers spoke of giving odds. The 2001 Kasparov-Chapman game was touted as an Odds match, not a handicap match. The same is true that the 2007 Rybka vs. Ehlvest (and later many others) match(es) were called Odds matches. Maybe the origin had something to do with chess being percieved as a games of chance? I really have no idea, but it might be fun to try to figure out.
Kasparov or his translator are clearly, if not wrong, at least on unsteady ground and using the term "piece" quite loosely and inaccurately.
Kasparov speaks English and is not making a mistake in terminology. He counts attackers and defenders to determine if an attack will work, and he counts pawns like pieces. In an attack that he determines will work, he will exchange an attacking minor piece for a defending pawn and consider it an even trade. I recommend watching his documentary as he comments on some of his best games and seeing his thinking is very informative. Below is a link to the first part and you should find the rest easily:
https://www.youtube.com/watch?v=BeEY66T0q80
Apples and oranges. That has absolutely nothing to do with anything here.