Why does renminbi have three currency units?

Yes, "fen" (分) is 0.01 yuan and is equivalent to "cent", and "jiao" (角) or "mao" (毛) is 0.1 yuan and is equivalent to "dime" in USD/CAD.

I guess whether something is a "unit" relates to its linguistic usage. A unit is something you would count in. For USD/CAD in English, you don't count in "dimes" e.g. "3 dimes", "4 dimes 2 cents", etc.; you would count in "cents" instead: "30 cents", "42 cents", etc. So "dime" isn't a unit, but "cent" is.

But for Renminbi in Chinese, you do count in "jiao"/"mao" for amounts that are at least 1 jiao. In verbal speech you would say "3 mao" or "4 mao 2" instead of "30 fen" or "42 fen". The banknotes and coins for 0.1 yuan and 0.5 yuan are labeled "1 jiao" and "5 jiao", not labeled in "fen". So "jiao"/"mao" is a unit in addition to "fen". (Well, coins below 1 jiao, counted in "fen", are rare now, but even when they were common, it was still true that you did count in "jiao"/"mao" for amounts at least 1 jiao.)